THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ON EVALUATION OF THE GRAPH TREES AND THE DRIVING POINT ADMITTANCE Noriyuki Nakagawa February, 1957 IP-204

ACKNOWLEDGEMENTS The writer wishes to express his gratitude to Professor Y. H. Ku, the thesis supervisor, for his advice, guidance and encouragement throughout this investigation. The writer also expresses his appreciation to Professor N. R. Scott, University of Michigan, for his constant encouragement and Professor J. W. Carr, III, for free use of the MIDAC computer.

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS...................... ii I. INTRODUCTION.................... 1 II. DEFINITIONS CONCERNING A GRAPH........... 3 III. ALGORITHM OF THE FOLDANT.............. 6 IV. THE TREE SUMMATION AND THE NUMBER OF TREES.... 8 V. DRIVING POINT ADMITTANCE.............. 14 VI. MIDAC PROGRAMMING FOR THE EVALUATION OF THE TREE SUMMATION..................... 16 A. The Number of Nodes n = 3.... 21 B. The Number of Nodes n = 4.. 21 C. The Number of Nodes n = 5.......... 21 D. The Number of Nodes n'= 6....... 22 E. The Number of Nodes n = 7.......... 24 VII. CONCLUSION..................o.. 39 VIII. REFERENCES..................... 40 iii

I. IRTRODUCTION Recently much interest has been aroused in topological treatment of networks. Kul called the attention of electrical engineers to original rules of Maxwell and Kirchhoff, which are the fundamentals of topological characteristics of the electrical network. So far few have 2 payed attention but Percival, who developed his theory from the graph of topology and did not, rather naturally, reach to such systematic and algebraic method a conventional matrix or determinant. Trent, on the other hand, showed how to calculate the characteristic numbers concerning the tree by making use of a "primitive node-pair connection matrix" but it is not as simple as Percival's. Syne6 investigated the matrix theory of electrical networks from the viewpoint of topology, Saltzer7 furthered the relationship between node analysis and mesh analysis from the sam point of view, and Reed systematized these approaches. Cederbra9'10 investigated fumdmental characteristics of the network determirlnt. These investigators did not try to find an efficient method of 11 obtaining characteristic numbers concerning the tree* Kron has been working on a large system but his approach to a small network was adopted by ynge.6 On the field of switching theory, however, Hohn proposed a simple, clear idea of a "connection matrix" and showed an interesting technique of the design of relay circuits. Why cannot we make use of the same matrix in the electrical circuit? Strangely enough, few have realized that it would still remain to be a powerful tool. Some work has been done in this thesis concerning this point. The purpose of this thesis, therefore, would deal with three subjects: the first is the introduction of a new algoritbhm of a -1

-2"Foldant"; the second to show that by making use of the foidant, such characteristic numbers concerning the tree as the total number, the tree summation of all possible trees of a network and the driving point admittance are performed systematically and algebraically without setting up a "primitive node-pair connection matrix", and the third to show that the algorithm of the foldant is so straightforward that the programming of the method may be realized. Throughout this paper, networks are supposed to be linear, time invariant and passive and to contain no mutual inductances. Although some of the restrictions can be removed, they are adopted for immediate convenience.

II. DEFINITIONS CONCERNING A GRAPH Several definitions are given below. Let us consider a graph of n nodes. The definition of a tree has been well accepted. For a matter of convenience, it will be rewritten below. Definition 1:5 A "tree" of a graph is a connected graph containing all the nodes of the graph and containing no loops. The following definition is the same as the one of a tree product in reference 5, but the present name is preferred for avoiding confusion in later development of the theory. Definition 2: The "tree designation" is the product of all branch designations of a tree. The following definition is set up for convenience of handling the later theories, although the idea has been used for years without definite nomenclature. Definition 3: The "tree summation" of a graph is the summation of all tree designations of a graph. The following two definitions of matrices are newly proposed here.3 These, however, will not be used directly in the development of our theory but only for deriving the two matrices given after them. Definition 4: The "primary branch matrix" (B) is an nxn upper diagonal matrix such that sum of the branch designations between nodes i and j, i < j (B) = (bij) =.1, i = j O. i >

-4Definition 5: The "primary branch number matrix" (N) is an nxn upper diagonal matrix such that number of the branches between nodes i and j, i < j (N) = (nij) =- i i = j 0, i > With the following two matrices, our theory will be developed. Definition 6: The "branch matrix" (B") is the (n-l)x(n-1) upper diagonal matrix derived from the primary branch matrix by deleting all diagonal entries of the latter. Thus the branch matrix has the following form: b b2 --- b b 12 1b3 bln-l ln 0 -- bn. b (B~) (1) o 0 0 bn —- 0ln Definition 7: The "branch number matrix" (N) is the (n-l)x (n-l) upper diagonal matrix derived from the primary branch number matrix by deleting all diagonal entries of the latter. A couple of examples will be given, concerning the definitions. Example 1. With branch designations A, B, and C of the network, Figure 1. Fg71~ Figure 1

a tree designation = AB the tree summation = AB + AC + BC the branch matrix = A C] 0 B the branch number matrix = o i1 Example 2. With branch designations A, A2, B and C of the network, Figure 2. 3 A2 Figure 2 a tree designation = A1B the tree summation = AB +A2B + A C + A2C + BC. A+ A2 Cl the branch matrix = l 0 B the branch number matrix = 2 1 0o 1

III. ALGORITHM OF THE FOLDANT The following neV algorithm is proposed here to make a direct computation of the tree surimation and the driving point admittance. Other application is anticipated. Let (A) be an nxn upper diagonal matrix. Then the "foldant" IAI is defined by the following recursive rules: IAI = Jall = all when n = 1, and a11 + a +a a +a, --- a +a 11 2n 2 2 + n a2 +an 1- a1n,n-1 nn O a22 a23 --- a2,n-1 Al = laijl = aln 0 0 3 nO 0 0 --- a a11 a12 a --- aln-1 O a22 + a3n a23 + a4n a2,n- + an + a2n 0 0 a33 3,n-1 O 0 0 --- a 151 12\ " 1n-l,n-1 all a12 a --- aln-l O a22 a23 --- a2'n-1 + a3n O 0 a33 + a4n -- a3,n-1 + ann 0 0 0 --- a n-l, n-6

+ + 0 0 0HO > rN H,I I I 1 I- o P H I I I I H3 H H H I N % % F-J 0 0:

IV. THE TREE SUMMATION AND THE NUMBER OF TREES Several theorems will be given. The following theorem 1 has 2 been proved as Theorem 6 in Percival's paper. Theorem 1: Let a graph be given with the tree summation T. Then, with reference to any branch designation b, T= To +b T' where T' is the tree summation of a graph derived from the given graph by deleting the branch b, and TI is the tree summation of a graph derived froml the given graph by identifying the nodes of the branch b. Proof: The tree sumration T can be partitioned into two terms, one a subset of those tree designations not containing the branch b and the other a subset of those tree designations containing the branch b. Let the first one be called T. Then it is clear that To is the tree summation of a graph derived from the given graph by deleting the branch b. I I i — -2 —— _ - -8 I 4 T, * I I Figure 3 Now, with. Figure 3, let us consider those tree suumrations containing branch b. Let the nodes of the branch b be 1 and n Let those nodes of the given graph, connected to the nodes I or n, be denoted as 2, 3,.., and i. And imagine a graph derived from the given graph by -8

-9deleting the nodes I and n and those brandcies which were connected to the nodes 1 or n from the nodes 2, 3,..., and i. Let the tree sumnmation of this graph be denoted as T. Then those tree summation containing the branch b of the given graph may be expressed as follows: i i b bl, T +b bb T onV o or V=2 4=2 i b X (bl1 + bn,)T #=2 ) (bl bn )To is certainly the tree summation T' of a graph derived -=2 from the given graph by identifying the nodes of the branch b, I and n. Thus the Theorem is proved. Corollary: Let a graph of n nodes be given with the tree summation T. Then, with reference to bin, the sum of the branch designations between nodes I and n, T = T1 + b T where T is the tree summation of a graph derived from the given graph by deleting all the branches between the nodes I and n, and T' is the tree summation of a graph derived from the given graph by identifying the nodes I and n. Proof: Let bin = b + b2 +.. + bj. By Theorem 1, T = T1 + b T1, where T~ is the tree summation of a graph derived from the given graph by deleting the branch bl. Similarly,

-10T = To + b T' 1 2 2 where T' is the tree summation of a graph derived from the given graph by deleting the branches bI and b2. Thus, T = To + (b + b2)T Repeating the process, T = T + (b + b2 +.b )T. + ) =T b T Thus the corollary is proved. By making use of the Corollary, the following new Theorem concerning the evaluation of the tree summation is derived. Theorem 2: The tree summation of a graph is equal to the foldant JB~I of the branch matrix (B~) of the graph. Proof: The proof is given by induction. When the number of nodes n = I, the graph is not defined. When n = 2, the theorem is obvious. Let the theorem be true when the number of nodes is (n-l). Now, let us consider a graph of n nodes with the tree summation T. Let the graph have the branch matrix (B~) of the equation (1) above. From the Corollary, T = T + bn T' I In I It is noticed that T' is a graph with (n-l) nodes. Hence the induction hypothesis can be applied to T'. Thus, b +b^ b +b --- b +b b2 + b2n b13 + b3n bl,n-,n 0 b23 b2i 25b, n-1 T = T1 + bin,(2) 0 0 --- b 2 n-2, n-l

-11where the graph with the tree summation T has ~{he following brancs matrix: b12 b13 bin-2 b1l 0 o b -,n-2 b n-1 0 b23 b2' n-2 2,n-1 2n O O --- 0 O b n-l,n Likewise, the Corollary is applied to the graph T, concerning the entry b. Repeating the process, T is developed into the form: 1i2 2n 1 bn,n-i+ bn-,n o b --- b T = bln 23 2,n-1 0 b23 + b3n --- b2 n-l,n + b^ 2 5n 2,n- n-,n O 0 --- b bl2 b1 bn+ bn23 2,n-l n-l,n 0 0 b O ~ O ^~ bn-2,n-l The right hand side is the foldant defined in Sec.II. Thus T = IB~I, and the Theorem is proved.

-12A Theorem concerning the enumeration of the number of trees will also newly be given as follows: Theorem 3: The number of trees of a graph is equal to the foldant of the branch number matrix of the graph. Proof: The proof follows the same procedure as the proof above. That is, it is clear from the Corollary that the number N of trees of a graph is expressed as I I N = N1 + N1, where N1 is the number of the trees contained in the tree summation T and N1 is the number of the trees contained in the tree summation T*. Then we can apply such induction procedure for the branch number matrix of the graph as used in Theorem 2. Thus Theorem 3 can be proved. A couple of examples will be given, illustrating Theorems 2 and 3. Example 3: On Figure 1, A C The tree summation = = C(A + B) + BA O B i 1 The nvtuber of trees = = 1.2 + 1.1 = 3 0 1 Example 4: With the branch designations A, B, C, D, E and F of the graph, Figure 4, C 4 3 F E Figure 4 Figure 4

-13ADF The tree summation = 0 E B 0 0 C 00C A+B D+C A D A D = F + B +C 0 E O E+C O E = F(D+C)(A+B+E) + FE(A+B) + BD(A+E+C) + B(E+C)A + CD (A+E) + CEA 111 22 11 11 The number of trees 0 1= + 001 01 02 01 = 2.3 + 1.2 + 1.3 + 2.1 + 3 = 16 The resut agrees ith those of Figure of Ku's paper. The result agrees with those of Figure 1 of Ku's paper.

V. DRIVING POINT ADMITTANCE As is shown in Ku's paper, the driving point admittance of a passive network without mutual inductance is obtained as B IB/An, where An is the sum of the products of y's taken (n-2) at a time, omitting all terms containing Yln or forming a closed circuit with it and all other terms forming closed circuits themselves. We have the following new theorem with respect to A: Theorem 4: Let a network of n nodes with the branch matrix of the equation (1) be given. Set the entry bn as 1. Then An is equal to the foldant appearing as the coefficient of bn in the equation (2) above Proof: It is seen from the derivation of the equation (2) that Y T' is the tree summation of the set of all trees containing Yln It is also seen from the rule for obtaining the driving point admittance that YlnAln is the tree summation of the set of all trees containing yln Hence the theorem is proved. A couple of examples will be given. Example 5: With the admittances A, B, and C of the network, Figure 5, 12B 4 10 ~ 2 A Figure 5 AQO A C0 0 0 = CA CB+ A -14

-15the driving point admittance at 14 ABC 1 = AB + BC + CA 1 1 1 A +B rC Example 6: With the admittances A, B, C, D, E and F of the network, Figure 6, 4F 4 3 Be ^ IE A 2 Figure 6 AADO A IBI = OEBO OOCO 0O O OOF 0 00F = FB A D + FAD 0 E+C O E = FBD(A+E+C) + FB(E+C)A + FCD(A+E) + FCEA A D F A+B C+D A D AD A^ = ] OEB= F +B +C OOC O E 0 E+C OE = F(C+D)(A+B+E) + FE(A+B) + BD(A+E+C) + B(E+C)A + CD(A+E) + CEA the driving point admittance at 15 FBD(A+E+C) + FB(E+C)A + FCD(A+E) + FCEA F(C+D(A+B+E) + FE(A+B) + BD(A+E+C) + BE+C)A + CD(A+E) + CEA The result agrees with if of Figure 1 of Ku's paper.

YV. MIDAC PROAMIM F TB EVALUATION1 TiE TRU St9MATI( The straightforward coding of the alphanumeric representation of the tree summation of any arbitrary graph having seven nodes or fewer was tried. If the number of node is eight, the number of total instructions of our progrmR would exceed the capaoity of the high-speed memory of C which is 512 as Described below. Furthermore, if the number of nodes is nine or more, even if we use the tape memory of MIAC, the result would be so bulky that it would hardly be worthwhile. That is, suppose we make a program so that each of the two term of the tree suination of the three-nodes graph shown in EXxaple 3 si printed on each row. Then it is known from Theorem 2 that the number of rowv of the four-nodes graph shown in ixaaple 4 is 3.2 3! - 6. Thus the number of rows of a nine-nodes graph with the branches connected between any nodes in any possible way would be 8' - 40,20*. It would need 611 aheets of paper to be printed, since 66 rows can be printed on a sheet of paper. As a ore elaborate progra, we may incorporwte Iron's method of tearing with ours, and the computation tiBe would be considerably reduced. owever, the number of result heets discussed above still would be too many for us to get aignificant iafortion. Thus it is conceivable that our way of the alphuaaxeric representation of the tree sunation is not appropriate for the analysis of a graph of many nodes. at we will aee what result we get A genera flow diagraf of the coding ia shown on Figure 7. Before going into ome detail of the codiag, a brief desoription will be given about the MIDAC computer. Patterned after SAC of the national Bureau of Btandard, MIIAC (Michigan Digital Automtic -i6

-17[START n07 7?-O NODES - NODES 5- NOgS I4-NODES 3-NODE $ QRAPH LRAPH RAPH RAPH GRAPH l^~ -~1~ _ Figure 7 General Flow Diagram of Coding Computer) is easentially a serial computer and has electric circuitry based on the so-oalled "dynaico flip-flop". The mercury acoustic delay storage has the capacity of 512 words. The drum systea has a capacity of 6,144 words. Etch word contains 44 binary digits plus a sign digit. The MZDAC speed is, on the average, 1000 ddition per second or 300 ltiplications per seon ed, with corresponding t es fo ther eoperations. The 45 binary dlgits of an instruction word are interpreted as consslted of three addresses and an operation code There are nineteen operations such as 1) Add, 2) Subtract,,) Multiply, Low Order, I) Multiply, High Order, 5) ultiply, Rounded, 6) Divide, 7) Power Extract,

-18-. 8) 8hift Number, 9) Extract, 10) Decimal to Binary Conversion, 11) Binary to Decimal Conversion, 12) Compare Numbers, 13) Compare Magnitude, 14) Base, 15) File, 16) Read In, 17) Read Out, 18) Move Tape Forward, and 19) Move Tape Backward. By making use of these instructlons, a detailed program can be prepared for the general flow diagram of Figure 7. At first, coding of a graph with three nodes will be explained. The result form of the Example 3 will be A.B (A+B)C. However, any case where either one or more of A, B, or C may be zero should be taken care of. Furthermore, in order that the routine can be used as a subroutine for a graph of more nodes, it is preferable that the coding can produce the following form A.B.WVlYX. (A+B)C.t.X.Y, where V is the entry decided by a four-nodes graph, W by a five-nodes graph, X by a six-nodes graph and Y by a seven-nodes graph. The flow diagram of a three-nodes grph is shown on Figure 8. In Figure 8, "Print" instruction is Qpecial subroutine, by which a content of a wort is checked five binary digits as a group and, if it is not zero, an alphabetic character corresponding to those five binary digits is read out into the output of Flexowriter as a printed form. Likewise, a flow diagram of a four-nodes graph is shown on Figure 9. On the figure, the symbol.- shows a ipecific subroutine, by which two or more alphabetic characters are put into a word meory so that theyray be read out by the subwutine -mentioned above in coneetion with Ftlgure 8.

-19-,l^I a l^ g? PRINT C; V I l I PRNIT BkV JUMP + q-v ~na yes> n l no E3 PRNIT SW no ^UMP PRINT a1 yes no r yes O Co? ~ EXT no CARRIAOG RETURN I n = n-I PRINT A.,, C. W Figure 8 Three-Nodes Graph Subroutine

-20A - BBO CE Co | DIV I GRAPH of 3-NODES SUBROUTINE CARRIAGE RETURN ~ E,? no IJUMP E~lol V ORAPH 3fT 4NODE SUBROUTINE I CARRIA6E RETURN DI ->Co lEi nlo GRAPH 9f 3- NODES SUBROUTINE Figure 9 Four-Nodes Graph Subroutine

-218imilar flow are drawn for graphs with more nodes and the result of running is shown in the next sections. A. The number of nodes n - 3 This is trivial and the result obtained agrees with Example 3 above. B. The number of nodes n - 4 The result obtained agrees with Example 4. C. The number of nodes n a 5 With branch designations, A, B, C, D, 1, F, G, H, I and J of the graph, Figure 10, the result obtained is presented below. The tree summation is the addition of the following 24 rows. (As for the symbols used, "." is stronger than "+" so that A.B+C.D+E, for instance, means A(B+C)(D+1).) A..B.D.G. M3J.C.D.G. A+I+D. C., 4 A.B.D+.G. A+I.C.D+G.H. /. A.3D.1..I. A4+<G..C.I.H.\3 A-4.B...H. / A+1B+. C+D+G. F. A.bM.D.I \ AM4l.C.D.I 1 ~ 2 A.~lD~.+O.I1 A+ D. C..14.x A+.b+l F.I Figure 10 +I+b+1M. O+D..I A+I.BD.J A+I+B.CO+.D.J A+I.3+D..J A+I+X+D..011 l.J A+I+B I.B.F+G.J A+I+FB. 0++4D.F+G.J

-22D. The number of nodes n = 6, Figure 11 A.B.D.O.K A+B.C.D.G.K A.B+D.E..K A+B+D. C E.GO.K A+3.B.F.G.K A+IB.O+D.F. G.K A.B.D+G.H.K A+B. C.D+.H.K A. B+D+ O..H.K A+B+D.O C E..H.K A+E.B.F.H.K A+E+B. ++. F. I \K A.B+H.D.I.K / I A+B+H. C.D.I.K A.B+H+D.+OG.I.K. A+B+H+D.C.+O.I.K / A44G.B+H.F.I.K A+BZ+GI4+. O+D.F I. )I A+I.BD.J.K /o,, A+I+B. C+H.D.J. A+I.B+D.E.J.K I A+I+B+D. C+H..J.K A+I+E.B.F O.J.K A+I++B. O+H+D.F+4.J.K, A.B.D.O+K.L 2 A+B. C.D.G.L A.B+D.E.G+K.L Figure 11 A+BD.C.E.G+K.L A+E.B. F. +K.L A+B+B. C+D. F. GA. L A.B.D+G+K.H.L. A+B. C..DK. H.L A.3BD. B+KCX..L A+B+D. OI.C.B..L A+E.B.F.H.L A++B. O+D+O+K. F. H. L A.B+H.D.I.L A+B+H.C.D.I.L A. B+H+D.E+G. I.L A+B+H+D. C. BG+..L A+EB++K.B+H.F.I.L A+4+G4+B+H. C+D.F. I.L A+I.B.D.J.L A+I+B. O+H.D.J.L A+I,B+D.B.J.L A+I+B+D. O+H.,. L A4+I+E.3.F+4.J.L A+I++B. 0+H4D.7K.J.L A.B.D+L.G.M. A+B. C.I+L.G.M

-23A.B+D+L.E.G.M A+B+D+L.C.E.G.M A+E.B.F.G.M A+E+B, C+D+L.F.G.M A.B.DL+.H+K.M A+B. C.DL+G.H+K.M A.B+D+L+.E.H+K.M A+B+D+L+G'. C.E.H+K.M A+E.B.F.H+K.M A+E+B. C+D+L+G.F.H+K. M A.B+H+K.D+L. I.M A+B++K CC.D+L.I.M A.B+H+K+D+L.E-G.. M A+B+H+K+D+L. C.E+G. I.M A+E+O. B+H+K.F. I.M A+E+G+B+H+. C+D+L. F. I.M A+I.B.D+L.J.M A+I+B. C+H+K.D+L.J.M A+I. BD+L.E.J.M A+I+B+D+L. C+H+K..E.J.M A+I+E.B.F+G.J.M A+I+E+B oC+H+K+D+L.F+G. J. M A.B+M.D.G.N A+B+. C.D. G.N A. B+MD.E+L.G.N A+B+M+D.C.I+L. G.N A+E+L.B+M.F.. N A+E+L+B+M. C+D. F. G. N A.B+M.D+G.H.N A+BB+M.C.D+G.H.N A. BM+D+G.I+L.H.N A+B+M+D+G. C.Z+L.H.N A+E+L.B+M.F. H.N A+E+J+BM. C+D+G. F.H.N A.B+M+H.D. I+K.N A+B+M+H.C.D.IK.N A B+M+H+D.E+L+G. I+K. N A+B+MH+D. C. E+L+G. I+K. N A+E++G.B+M+H.F.I+K.N A+E+J+G+B+M+J. C+D. F. IK.N A+I+K. B+.D.J.N A+I+K+B+M. C+H.D.J.N A+I+K. B+M+D.E+L.J.N A+I+K+B+MD. C+H.E+L.J.N A+I+K+E+L3. B+M. F+G. J.N A+I+K+1+L+B+M. C+H+D. F+G.J.N A+N.B.D.G.O A+N+B. C+M.D.G.O A+N.B+D.E.G.0O A+N+B+D. C+M M.E.O. A+N+B.B.F+L G..0 A+N-++B. C++D. F+L. G. 0 A+N.B.D+G.H.O

-24A+N+B. CM.D+G.H. A+N. B++G.E.H.0 A+N+B+D+. C+M. E. H. A+N+E.B.F+L.H. O A+N+B+B. C+M+D+G.+L.H.0 A+N. B+H.D. I.0 A+N+B+H.C+M.D I. 0 A+N.B+H+D.B+. I. A+N+B+H+D.C+M.E+.I. 0 A+N+E+G. B+H.F+L 1. 0 A+N++G++H. C+M+D.F+L. I.0 O A+N+.B.D.J+K.0 A+N+I+B. C4+H.D.J+4. 0 A+N+I.B+D.I.J+K.0 A+N+I+B+D. C+M+H.B.JK. 0 A+N+I+. B. F+L+O.J+K.0 A+NI+I++B. CM+lH+D.F+L+G. J+K.O E. The number of nodes n = 7, Figure 12 A.B.D.G.K.P A+B.C.D.G.K.P A.B+D.I.G.K.P A+B+D.C.E.G.K.P A+I.B.F.G.K.P A+E+B.C+D.F. G.K.P K A.B.D+G.H.K.P L L A+B.C.D+G.H.K.P / A.B+D+G.E.H.K.P/ A+B+D+G.C.E.H.K.P\ A+E.B.F.H.K.P 7 \ \ A+3+B.C+D+O.F.H.K.P A.B+H.D.I.K.P A+B+H.C.D.I.K.P /\ A.B+H+D.B +G.I.K.P A+B+H.CD. +G.. I.K.P / A+E+G.B+H.F.l.K.P A+IG+B+. C+D.F. I.K.P A+I..B.D.J.K.P A+I+B.C+H.D.J.K.P A+IB+D,..J.KI.P A+I+B+D.C+H.E.J.K.P 2 A+I+.B.F4+.J.K.P A+I4+B. C+B+D. F+G.J K. P A.B.D.S+K.L.P A+B.C.D.O+K.L.P A.B+D...C+K.L.P Figure 12

-25A+B+D.C.E. G+K.L.P A+E.B.F.G+K.L.P A+E+B. C+D. F. G+K L. P A.B.D+G+K.H.L.P A+B.C. D+G+K.H. L.P A.B+D+G+K.E.H.L.P A+B+D+G+K.C.E. H.L. P A+E.B. F H.L.P A+E+B. C+D+G+K. F.. L. P A.B+H.D. I'.L.P A+B+H.C.D. I L. P A.B+H+D.. +G+K.I. L.P A+B+H+D.C. ++K. I. L. P A+E+G+K.B+H.F.I.L.P A+E+G+K+B+H. C+D. F. I. L. P A+I.B.D.J.L.P A+I+B.C+H.D.J.L.P A+I.BD.E.J.L.P A+I+B+D. C+H.E.J. L. P A+I+E.B.F+G+K.J.L.P A+I+E+B. C+H+D.F++K. J. L. P A.B.D+L..M.P A+B. C.D+L..M. P A.B+D+L.E.G.M.P A+B+D+L. C.E.G.M.P A+E.B.F.G.M.P A+E+B. C+D+L. F. G.M. P A.B.D+L+G.HK.M.P A+B. C.D+L+.H+K.M.P A.B+D+L+.B.HI+K.M.P A+B+D+L+G. C.I.H+K. M. P A+E.B.F.H+K.M.P A+E+B.C+D+L.F.H+K.M.P A.B+H+K.D+L.I.M.P A+B+H+K. C.D+L. I.M.P A.B+H+K+D+L.E+..M.P A+B+H+K+D+L. C.E+. I.M.P A+E+G. B+H+K.F. I.M.P A+E+G+B+H+K. C+D+L. F. I.M.P A+I.B.D+L.J.M.P A+I+B. C+H+K.DL.J.M.P A+I.B+D+L.E.J.M.P A+I+B+D+L. C+H+K..J.M. P A+I+E.B.F+G.J.M.P A+I+E+B. C+H+K+D+L. F+.J.M. P A.B+M.D.G.N.P A+B+M. C.D.O.N.P A. B+M+D.+L.'G...P A+B+*D oQ.k.L.G.N.P A+I+L.3B.. F. G.N.P A+E+L+BM. C+D..F. G.N. P A.B+M.D+O.H.N.P A+BM. C.D+.H.N.P

-26A.B+M+D+.E+L.H.N.P A+B+MD+G.C+E+L.H.N.P A+E+L.B+M.F.H.N.P A+E+L+B+M. C+D+G. F.H.N. P A.B+M+H.D.I+K.N.P A+B+M+H. C.D. I+K. N. P A.B+M+H+D.X+L+G. IK.N.P A+B+M+H+D. C.E+L+G.I+K.N.P A+E+L+G. B+M+H.F. I+K.N.P A+E+L+G+B+M+H. C+D. F. I+K. N. P A+I+K.B+M.D.J.N.P A+I+K+B+M. C+H.D.J.N.P A+I+K.B+M+D.E+L.J.N.P A+I+K+B+M+D. C+H E+L. J. N. P A+I+K+E+L.B+M.F+G.J.N.P A+I+K++L+B+M. C+H+D. F+G.J.N. P A+N.B.D.G.O.P A+N+B. C+M. D. G. 0. P A+N. BD.E.G.0.P A+N+B+D.C+M. E.G.O.P A+N+E.B.F+L.G.0.P A+N+E+B. C+M+D. F+L. G. 0. P A+N.B.D+G.H. O.P A+N+B. C+M.D+G.H.O. P A+N.B+D+G.E.H.O.P A+N+B+D+G. C+M.E H. Q.P A+N+E.B.F+L.H. O. P A+N+E+B. C+M+D+G. F+L. H. O. P A+N.B+H.D.I. O.P A+N+B+H. C+M.D. I.O. P A+N. B+H+D.E+G. I.. P A+R+B+H+D. C+M.BE+G.0O. P A+N+E+G B+H.F+L. 1.0.P A+N+E+G+B+H. C+M+D. F+L. I. O. P A+N+I.B.D.J+K.0.P A+N+I+B. C+M+H. D.J+K. O. P A+N+I.B+D.E.J+K O. P A+N+I+B+D. C+M+H.E.J+K. 0. P A+N+I+E. B. F+L+G. JK, 0. P A+N+I+E+B. C+M+H+D.F+L+G. J+K. O. P A.B.D.G.K+P. A+B.C.D.G.K+P. A.B+D..G.K+P. A+B+D.C.E.G.K+P.Q A+E.B.F.G.K+P.Q A+E+B.. O+D. F.G.+P. Q A.B.D+G.H.K+P.Q A+B. C.D+G.H.K+P. Q A.B+D+.E.H.K+P.Q A+B+D+G. C.E.H.K+P. Q A+E.B.F.K+P. Q A+E+B. C+D+G.F.K+P. Q

-27A.B+H.D. IK+P.Q A+B+H.C.D.I.K+P.Q A.B+H+D.E+G.I.K+P.Q A+B+H+D. C.E+G. I.K+P.Q A+E+G. B+H.F.I.K+P.Q A+E+G+B+H. C+D. F. I.K+P. Q A+I.B.D.J.K+P. A+I+B. C+H. D.J.K+P. Q A+.B+D.E.J.K+P.Q A+I+B+D.C+HE. J.K+P. Q A+I+E.B.F+G.J.K+P.Q A+I+3+B. C+H+D.F+G.J.K+P.Q A.B.D.G+K+P.L.Q A+B. C. D. G+K+P. L. Q A.B+D..G+K+P.L.Q A+B+D.C.E.G+K+P.L.Q A+E.B.F.G+K+P. L.Q A+E+B. C+D. F. G++P. L.Q A.B.D+G+K+P.H.L.Q A+B.C D+G+K+P.H.L.Q A.B+D+G+K+P.E.H.L. A+B+D+G+K+P. C.E.H. L.Q A+E.B.F.H.L.Q A+E+B. C+D+G+K+P. F.. L. Q A.B+H.D.I.L.Q A+B+H.C.D. I. L.Q A.B+H+D.E++K+P.I.L.Q A+B+H+D.C.E+G+K+P. I. L. Q A+E+G+K+P. B+H.F. I. L. Q A++G+K+P+B+H. C+D. F.I. L.Q A+I.B.D.J.L.Q A+I+B. C+H.D.J.L.Q A+I.B+D.E.J.L.Q A+I+B+D.C+H.E.J.L.Q A+I+.B.F++K+P.J. L.Q A+ +E+B. C+H+D. F++K+P.J. L Q A.BD+L. G.M.Q A+B.C.D+L.G.M.Q A.B+D+L.E.G.M.Q A+B+D+L.C.E. G.M. Q A+E.B.F.G.M.Q A++B.C+D+L. F.G.M.Q A.B.D+L+G.H+K+P.M.Q A+B. C.D+L+G.H+K+P.M.Q A.B+Dt+L+G.E.H+K+P.M.Q A+B+D+L+G.C.E.H+K+P.M.Q A+E.B. F. H+K+P.M.Q A+E+B. C+D+I+. F.H+K+P. M.Q A. B+HL+P.DL. I.M.Q A+B+H+K+P.C.D+L.I.M.Q A.B+H+K+P+D+L.E+G.I.M.Q A+B+H++P+D+L.C.EO. I.M. Q

-28A+E+G. B+H+K+P...I.M Q A+E+G+B+H+K+P.C+D+L.F. I.M.Q A+I.B.D+L.J.M.Q A+I+B. C+H+K+P.D+L.J.M. - A+I.B+D+L.E.J.M.Q A+I+B+D+L. C+H+K+P.E.J.J A+I+E.B..F+.J.M.Q A+I+E+B. C+H+K+P+D+L. F+G.J.M.Q A.B+M.D.G.N.Q A+B+M.C.D.G.N.Q A.B+M+D.E+L..N.Q A+B+MD.C.+L.G.N Q A+E+L. B+d..G.N.Q A+E+L+B+M.C+D F.G.N.Q A.B+M.D+G.H.N.Q A+B+M.C.D+G.H.N.Q A.B+M+D+G.E+L.H.N.Q A+B+M+D+G.C.E+L.H.N.Q A+E+L.B+M.F.H.N. Q A+E+L+B+M. C+D+G. F.H.N.Q A.B+M+H.D. I+K+P.N.Q A+B+M+H. C.D. I+K+P.N.Q A.B+4M+H+D.E+L+G. I+K+P.N.Q A+B+M++D. C.E+L+G I+K+P..N Q A+E+L+G. BM+H.F. I+K+P.N.Q A+E+L+G+B+M+H. C+D. F. I+K+P.N Q A+I+K+P.B+M.D.J.N.Q A+I+K+P+B+M. C+H.D.J. N.Q A+I+K+P.B+M+D.E+L.J.N.Q A+I+K+P+B+M+D. C+H.B+L. J..Q A+I+K+P++L. B+M. F+G.J J.N Q A+I+K+PB++L+B+M. C+H+D.F+G.J.. Q A+N.B.D.G.O.Q A+N+B. C+M.D.G.O. Q A+N. B+D.E.G..Q A+N+B+D. C+.E.0O.Q A+N+E.B.F+L.G.0.Q A+N+E+B. C++D. F+L. G. 0. Q A+N.B.D+G.H.O.Q A+N+B. CM.D+G.H.0.Q A+N.B+D+G.B.H.O.Q A+N+B+D+G.C+M.E.H.O.Q A+NE.B.F+L.H.O. Q A+N+E+B. C++D+G. F+L. H. 0.Q A+N.B+H.D.I. O.Q A+N+B+H. C+MD. DI. 0. Q A+N. B+H+D.1+..I..Q A+N+B+H+D. C+M.M G. I. O. Q A+N+E+0. B+.F+L.I.0.Q A+N+B+G+B+H.C+M+D.F+L.I O. Q A+N+I.B.D.JK+P.0.Q A+N+I+B. C^++H.D.J+K+P. O. Q A+N+I.B+D.E.J+K+P.O.Q A+N+I+B+D. C+H...J+K+P.0.Q A+N+I+E.B.F+I.J+K+P. O. Q

-29A+N+I+E+B. C+M+H+D. F+L+G. J+K+P.. Q A.B.D.G+Q.K.R A+B. C.D. G+Q.K.R A.B+D.E.G+Q.K.R A+B+D.C.E. G+Q.K.R A+E.B.F.G+Q.K.R A+E+B. C+D.F. G+Q.K.R A.B.D+G+Q.H K.R A+B.C.D+G+Q.H.K.R A.B+D+G+Q.E.HK.R A+B+D+G+Q. C. E. H.K. R A+E.B.F.H.K.R A+E+B. C+D+G+Q. F. H.K.R A.B+H.D.I.K.R A+B+H. C.D.I.K.R A. B+H+D. E+G+Q. I.K.R A+B+H+D C. E+G+Q I..K.R A+E+G+Q.B+H.F.I.K.R A+E+G+Q+B+H. C+D. F. I.K.R A+I.B.D.J.K.R A+I+B.C+H.D.J.K.R A+I.B+D.E.J.K.R A+I+B+D. C+H.E. J.K.R A+I+E.B.F+G+Q.J.K.R A+I+E+B. C+H+D. F+G+Q. J.K.R A.B.D.G++K.L+P.R A+B. C.D.G++K.L+P.R A.B+D.E.G++K.L+P.R A+B+D. C.E.G+Q+K. L+P.R A+E.B.F.G+Q+K.L+P.R A+E+B. C+D. F. G+Q+K. L+P.R A.B.D+G+Q.+K.H.L+P.R A+B. C.D+G+Q+K.H. L+P.R A.B+D+G+Q+K.E.H.L+P.R A+B+D+G+Q+K. C.E. H. L+P.R A+E.B.F.H.L+P.R A+E+B. C+D+G+Q+K. F. H. L+P.R A.B+H.D.I.L+P.R A+B+H.C.D.I.L+P.R A.B+H+D.E+G+K. I.L+P.R A+B+H+D. C.E+G+Q+K. I. L+P.R A+E+G+Q+K. B+H.F.I. L+P.R A+E+GQ+K+B+H. C+D. F,1. L+P.R A+I.B.D. J. L+P.R A+I+B.C+H.D.J. L+P.R A+I.B+D.E.J. L+P.R A+I+B+D. C+H.E.J. L+P.R A+I+E. B.F+G+Q+K.J. L+P.R A+I+E+B. C+H+D. F+G+Q+K.J. L+P.R A.B.D+L+P.G+Q.M.R A+B. C.D+L+P.G+Q.M.R A.B+D+L+P.E. G+Q.M.R A+B+D+L+P. C. E. G+. M.R A+E.B.F.G+Q.M.R A+E+B. C+D+L+P. F. F+Q.M.R

-30A. B. D+L+P+G+Q. H+K.M.R A+B.C.D+L+P+G+Q.H+K..R A. B+D+L+P+G+Q. E.H+K.M.R A+B+D+L+P+G+. C..H+K. M.R A+E.B.F.H+.MJ.R A+E+B. C+D+L+P+GQ..H+K.M.R A.B+H+K.D+L+P.I.M.R A+B+H+K. C. D+L+P. I.I.R A. B+H+K+D+L+P.BG+Q. I.M.R A+B+H+K+D+L+P. C.E++. I.M.R A+E+G+Q. B+H+K. F. I.M.R A+E+G+QB+H+K. C+D+L+P F.I..R A+I.B.D+L+P.J.M.R A+I+B. C+H+K. D+L+P.J.M.R A+I.B+D+L+P.E.J.M.R A+I+B+D+L+P. C++K. E.J.M.R A+I+E.B. F+G4.J.M.R A+I+E+B. C+H+K+D+L+P.F4+. J..R A. B+M.D.G+Q..R A+B+M. C.D. Q. N.R A.B+4D.ZEL+4P. 4Q. N.R A+B+M+D. C.E+L+P. G+..R A+E+L+P. B+M. F.G+Q..R A+E+L+P+B+M. C+D. F. GQ. N.R A.B+M.D+G+Q.H.N.R A+B+. C.D+G+Q.H.N.R A.B+M+D..E+L+P.H.N.R A+B+M+D+G+Q. C.E+L+P.H.N.R A+E+L+P.BM. F.H.N.R A+4EL+P+B+M. C+D4..H.N.R A. B++H.D.I+K.N.R A+B++H.C.D.4I+.1.R A.B+4+H+D.E+L+P+G+Q.IK.N.R A+B+M+H+D. C B+L+P+O+Q. I+.N.R A+E+L+P+OQ. B+M+H.F.IK.N.R A+E+L+P+Q+B++H. C+D.F. IK. N.R A+I+.B+M.D.J.N.R A+I+KB+M. C+H.D.J.N.R A+I+K. B+D. E+L+P.J.N.R A+I+K+B+M+D. C+H. +L+P.J J.RN A+I+K+E+L+P. B+. F+G+.J.N.R A+I+K+E+L+P+B+M. C+I+D.F404.J.,RR A+N.B.D.G+Q.O.R A+N+B. C+M.D G^. O.R A+N.B+D.E.0 G.O.R A+N+B+D.C+M. E.Q,0.R A+N+.B.F+L+P. GQ.O.R A+N+E+B.C C++D. F+4LP. G+Q.0.R AN.B.D+Q.H.0.R A+N+B.C+MJ.D+-.Q H.0.R A+N.BB+D+Q+Q. E.0.R A+1+B*4D+. C+M.. H. O.R A+N+.B.F+LP. H.O.R

-31A+N+E+B. C+4D+OQ. F+L+P.H. 0.R A+N.B+H.D.I.0.R A+N+B+H.C+M.D.I.0.R A+N.B+H+D.E+H. I. 0.R A+N+B+H+D. C+M. E4. I 0.R A+N+lO+Q.B+H. F+L+P..0.R A+N+E+G+Q+B+H. C+M+D. F++P. I. 0.R A+N+I.B.D.J+K. O.R A+N+I+B. C++H.D.J+K.0.R A+N+I.B+D.3.J+K..R A+N+I+B+D. 4+M+H. E.J+K.0.R A+N+I+Z.B. F+L+PQ. J+K.0.R A+N+I+E+B. C+4+H+D. F+L+P+G+Q.J+K.0.R A.B.D+R.G.K.8 A+B.C.D+R.G.K.8 A.B+D+R.E.G.K.S A+B+D+R. C.E.G.K. A+E.B.F.G.K.S A++B.C+D+R.F.G.K.S A.B.+R+.H+Q.K.S A+B.C.D+R+G.H+Q.K.S A.B+D+R+G.E.H+Q...S A+B+D+R+G. C..H+.K.8 A+E.B.F.H+Q.K.8S A+E+B. C+D+R+G. F. H+Q.K.8 A.B+H+Q.D+R.I.K.8 A+B+H+Q..D+R.I.K. A.B+H+Q+D+R.Z+O.I.K.8 A+B+H+QD+R.C+.0C.l.K.8 A+E+G.B+H+Q.F. I.K. A+E+G+B+HQ. C++R.F. I.K. A+I.B.D+R.J.K.S A+I+B. C+H +.D+R.J.K S A+I.B+D+R.E.J.K.S A+I+B+D+R,C+H+Q.E.J.K8 A+I+E.B.F+.J.K.8 A+I+E+B. C+H+Q+.D F+.J.K. S A.B.D+R.G+K.L.S A+B. C.D+R..L.8 A.B+D+R.E.G+K.L.S A+B+D.+. C.E.G+K. L.8 A+E.B.F. +K.L.8 A+E+B. C+D+R. F. G+. L. S A.B.D+R+G+K.H+Q.L.S A+B.C.D4R++K.H+Q.L.8 A.B+D+R+G+K.E.H+QL.S A+B+D+R++K. C.E.H+Q.L. S A+E.B.F.H+Q. L.S A+E+B. C+D+R+K.F.H+Q.L. S

-32A,.B+H+QD+R.I.L.S A+B+H+Q.C.D+R.I.L.S A. B+H+Q+D+R. E+G+K. I. L.S A+B+H+Q+D+R. C.E+G+K. I. L, S A+E+G+K. B+H+Q.F.I. L. A+E+G+K+B+H+Q. C+D+R F I.,L. S A+I.B.D+R.J.L.S A+I+B. C+H+Q.D+R.J.L.S A+I.B+D+R.E.JL.S A+I+B+D+R.C+H+ Q.E.J L. S A+I+E.B. F+G+K.J.L.S A+I+E+B. C+H+D+Q+R. F+G+K J. L. S A.B. D+R+L.G.M+P.S A+B. C.D+R+L. G.M+P.S A.B+D+R+L.E. G.M+P. S A+B+D+R+L. C.E.G.M+P. S A+E.B.F.G.M+P.S A+E+B.C+D+R+L. F. G.M+P. S A.B.D+R+L+G.H+Q+K.M+P.S A+B. C. D+R+L+G. H+Q+K. M+P S A. B+D+R+L+G. E. H++K. M+P. S A+B+D+R+L+G. C E. H+Q+K. M+P. S A+E.B.F. H+Q+K.M+P.S A+E+B. C+D+R+L+G. F H+Q+K. M+P. S A.B+H+Q+K.D+R+L. I.M+P. S A+B+H+Q+K. C. D+R+L. I.M+P. S A. B+H+QK+D+R+L.E+G. I. M+P. S A+B+H++K+D+R+L. C.E+..M+P.8 A+E+G. B+H+QK. F. I.M+P. 8 A+E+0+B+H+QK. C+D+R+L. F.. M+P. S A+I.B.D+R+L.J.M+P.S A+I+B. C+H+Q+K.D++L.J.M+P. S A+I.B+D+R+L.E.J.M+P. S A+I+B+D+R+L. C+H++K.E.J.+P. S A+I+E.B.F+G.J.M+P.S A+I+E+B. C++H+Q+K+D+R+L. F+G. J.M+P. S A.B+M+P. D+R.G. N. S A+B+M+P.C.D+R G.N. A.B+M+P+D+R.E+L. G.N. S A+B+M+P+D+R.C.E+L. G. N. S A+E+L. B+M+P.F. G. N. S A+E+L+B+M+P. C+D+R. F. G. N. S A.B+M+P.D+R+G.H+.N.S A+B+M+P. C, D+R+G. H+Q. N. S A.B+M+P+D+R+G. E+L.H+Q.N.S A+B+M+P+D+R+G. C.E+L.H+Q.N. S A+E+L. B+M+P. F. H+Q. N. S A+E+L+B+M+P. C+D+R+G. F.H+Q.N. S A.B+M+P+H+Q.D+R.I+K.N. A+B+M+P+H+Q. C. D+R. I+K.i.N. S A. B+M+P+H+D+R. E+L+G. I+K. N. S

-33A+B+M+P+H+Q+D+R. C. E+L+G. I+K N. S A+E+L+G. B+M+P+H+Q. F. I+K. N. S A+E+L+G+B+M+P+H+Q. C+D+R. F. I+K.N. S A+I+K. B+M+P. D+R.J.N.S A+I+K+B+M+P. C+H+Q.D+R.J.N. S A+I+K,B+M+P+D+R.E+L. J.N.S A+I+K+B+M+P+D+R. C+H+Q.E+L.J.. S A+I+K+E+L. B+M+P. F+G, J. N. 8 A+I+K+E+L+B+M+P. C+H+Q+D+R. F+G. J. N. S A+N.B.D+R.G.0.S A+N+B.C+M+P.D+R.G.0.S A+N. B+D+R.E. G. 0. A+N+B+D+R.C+M+P. E. G.0 S A+N+E.B F+L. G. 0. S A+N+E+B. CM+P+D+R. F+L. G. O. S A+N.B.D+R+G.H+.0.8 A+N+B. C+M+P D+R+G.H+. 0. S A+N.B+D+R+G.E.H+Q.O. A+N+B+D+R+G. C+M+P.E.H+Q. O. S A+N+E.B. F+L.H+Q. 0. S A+N+E+B. C+M+P+D+R+G. F+L. H+Q..S A+N.B+H+Q. D+R.1.0.S A+N+B+H+Q. C+M+P. D+R.I. 0. S A+N. B+H+QD+R. E+G. I. 0. S A+N+B+H+Q+D+R. C+M+P. E+G. I. O. S A+N+E+G. B+H+. F+L. 1.0. S A+N+E+G+B+H+Q. C+M+P+D+R. F+L. I.. S A+N+I.B.D+R.J+K.O.S A+N+I+B. C+M+P+H+Q.D+R.J+K. 0. S A+N+I.B+D+R.E.J+K..S A+N+I+B+D+R. C+P+H+Q.. J+K. O.8 A+N+I+E. B. F+L+G.J+K.0.8 A+N+I+E+B. C+P+H+Q+D+R. F+L.J+K. O. S A.B+S.D.G.K.T A+B+S. C. D. G.K.T A.B+S+D.E+R.G.K.T A+B+S+D.C.E+R. G.K.T A+E+R.B+S.F.G.K.T A+E+R+B+S. C+D. F. G. K. T A.B+S.D+G.H.K.T A+B+S. C.D+G.H.K.T A.B+S+D+G.E+R.H.K.T A+B+S+D+G. C.E+R.H.K.T A+E+R.B+S.F.H.K.T, A+E+R+B+S. C+D+G. F.H.K.T A.B+S+H.D.I+Q.K.T A+B+S+H. C.D. I+Q.K.T A.B+S+H+D.E+R+G. I+Q.K.T A+B+S+H+D. C. E+R+G. I+Q,K. T A+E+R+G.B+S+H.F.I +QK.T A+E+R+G+B+S+H. C+D. F. I+Q.K. T A+I+Q.B+S.D.J.K.T A+I+Q+B+S. C+H.D.J.K.T

-34A+I+Q.B+S+D.E+R.J.K.T A+I+Q+B+S+D. C+H.E+R.J.K.T A+I++E+R.B+S.F+G.J.K.T A+I+Q+E+R+B+S. C+H+D.F+G.J.K.T A.B+S.D.G+K.L.T A+B+S. C.D.G+ K.L.T A.B+S+D.E+R.G+K.L.T AiB+S+D. C.E+R.G+K.L.T A+E+R. B+S. F. G+K. L. T A+E+R+B+S. C+D. F. G+K.L.T A.B+S.D+G+K.H.L.T A+B+S. C. D++K. H. L.T A.B+S+D+G+K.E+R.H.L.T A+B+S+D+G+K.C.E+R.H.L.T A+E+R.B+S.F.H. L.T A+E+R+B+S.C+D+GK. F.H.L.T A.B+S+H.D.I+Q.L.T A+B+S+H. C. D. I+Q. L. T A.B+S+H+DD.E+R+G+K.I+Q.L.T A+B+S+H+D. C. E+R+G+K. I. L. T A+E+R+G+K. B+S+H. F. I+Q. L. T A+E+R+G+K+B+S+H. C+D. F. I+Q. L.T A+IQ. B+S.D.J.L.T A+I+Q+B+S. C+H.D.J.L. T A+I+Q.B+S+D.E+R.J.L.T A+I+Q+B+S+D. C+H.E+R.J.L.T A+I+Q+E+R. B+S. F+G+K.J. L.T A+I+Q+E+R+B+S. C+H+D. F++K.J.L.T A.B+S.D+L.G.M.T A+B+S. C.D+L.G.M.T A. B+S+D+L.E+R. G.M.T A+B+S+D+L. C.E+R. G.M. T A+E+R.B+S.F.G.M.T A+E+R+B+S. C+D+L. F.G.M.T A.B+S.D+L+G.H+K.M.T A+B+S. C.D+L+G.H+K.M.T A. B+S+D+L+G. E+R. H+K. M.T A+B+S+D+L+G.C.E+R.H+K. M. T A+E+R.B+S.F.H+K.M.T A+E+R+B+S. CD+L+G. F.H+K.M. T A. B+S+H+K.D+L. I+Q.M.T A+B+S+H+K. C. D+L. I+Q. M.T A.B+S+H+K+D+L. E+R+G. I+Q.M.T A+B+S+H+K+D+L. C.E+-R+G. Iq.M. T A+E+R+G. B+S+H+K. F. I+Q.M. T A+E+R+G+B+S+H+K. C+D+L. F.I+Q. M.T A+I+Q.B+S.D+L.J.M.T A+I+Q+B+S. C+H+K.D+L.J.M. T A+I+Q. B+S+D+L,E+R.J.M.T A+I+Q+B+S+D+L. C+H+K.E+R.J M.T A+I+Q~E+R.B+S.F+G.J.M.T A+I+Q+E+R+B+S. C+H+K+D+L. F+G. J. M.T A.B+S+M.D. G.N+P.T

-35A+B+8+M.C.D..N+P. T A. B++M+D.E+R+L. G. N+P. T A+B+S+M+D. C. EI+L...N+P.T A+3+R+L.B++. F.G.N+P.T A+E+R+L+B+t M. C+D. F.. N+P. T A. B+S+M.D+G. HN+P. T A+B++M. C.D+G.H.N+P.T A.B+S+M+D+G. E+R+L..N+P. T A+B+84+D+G. C..B+R+L..N+P. T A+ZE+R+L. B++M. F.H. N+P. T A++R+L+B+S+M. C+D+.F.H. N+P. T A.B+8M+H.D.+Q+K.N+P.T A+B+8+M+H. C D..+Q+K. N+P.T A.,4B4+ ++D.B++L.I.1N I N+P.T A-,,-W.D,. C.E+R+L+.I++.N+P. T A++R+L4O. B+8++H. F. I+QK. N+P. T A4.+L,+G+3B+S+M+. C+D.r. I-Q+K.N+P.T A+I+Q+K.B+S+M.D.J.N+P.T A+I4QK+B+84M. C+H.D.J.J+P.T A+IK. B+8+M+D.OR I +R+L. J. N+P. T A+I+Q+K+B++M+D.C+H.E+R+L. J.N+P. T A+IQ+K+3+R+L. B++M. F+G.J.N+P.T A+I+Q+K+3+R+L+B+8+M. C+H+D. F+G.. N+P. T AN+P. B.D.G.O.T A^++P+3+. C+.D DO.O.T A+N+P.3+S+D.1R.G. O.T A+N+P+B+8D.C+M. +R., O.T A+N+P+E+R.,l8.F+L.G.O.T A+N+P+E+R+B+. C+M+D. F+L. G. 0. T A+N+P.B+8.D+G0.O. T A+N+P+B+S. C+M.D+G.H.O.T A+i+P.B+8+DO,E+R.1.O.T A+N+P+B++D+. C+M. E+R. H. O. T A+N+P+E+R.+S. F+L.L.O.T A+N+P+E4+S+8.C-M+D+. F+L..O. T A+N+P. B++H. D. I+Q. O.T A+N+P++S+. C+ D. I+Q..T A+N+P. B++H+D.E+R+0. I+Q. O. T A+N+P+B+8+N+D. C+M.C+R+0. I+Q. O0 T A+N+P+BER+G, +S+H. F+L. I+Q. O. T A+N+P+1+R+G+BH++H. C+M+D. F+L.I+QO. T A+N+P+I+4.+8.D.J+K.O.T A+N+P+I+4++. C+M+.D.J+K.O T A+N+P+Is+Ql.B++D.3+R.J+K. O. T A+N+P+I+QB+8+D. C+M+H.E+R. J+K. O.T A+NtP+I+Q+E+R.B+8. F+L+. J+K. 0.T A+N+P+I+Q++R+B+8. C+M++D. F+L+G.J+K. O. T A+T.B.D.K.U A+T+B. O+.D.O.K. A+T.D.3E.OE.K.U A+T+B+D.4.B.OG.K.U A+T+B.B.FR.a.K.U A+TB++B.C+8+D.F+R.. K.U

-36A+T.B.D+G.H.K.U A+T+B.C+S.D+G.H.K.U A+T.B+D+G.E.H.K.U A+T+B+D+G. C+S E.H.K.U A+T+E.B.F+R.H.K.U A+T+E+B. C+S+DG. F+R.1.K.U A+T.B+H.D.I.K.U A+T+B+H. C+8.D. I.K. U A+T. B+H+D. E+G. I K.U A+T+B+H+D. C+8.E+G. I.K.U A+T+E+.B+H.F+R I.K.U A+T+E+G+B+H. C+8+D.F+R..K.U A+T+I.B.D.J+Q.K.U A+T+I+B. C+S+H.D.J+Q..U A+T+I.B+D.E.J+Q.K.U A+T+I+B+D. C+S+H..J+Q.K. U A+T+I+E.B. F+R4.J+Q.K.U A+T+I+E+B. C+8+H+D. F+R+G.J+.K.U A+T.B.D.G+K.L.U A+T+B.C+S.D.G+K. L.U A+T.B+D.E.G+K. L.U A+T+B+D. C+S.E. G+K. L.U A+T+E. B.F+R. G+K.L.U A+T+E+B. C+S+D. F+R. G+K. L. U A+T.B. D+G+K.H.L.U A+T+B. C+. D+G+K.H. L.U A+T.B+D+G+K..H.L.U A+T+B+D+ GK.C+S.E.H...U A+T+E.B.F +.H. L.U A+T++3B.C+8+DG+K.. +H.L.U A+T.B+H.D.I.L.U A++IB+H. +8s.D I.L.U A+T. B+H+D. ++K. I. L.U A+T+B+l+D. C+8.1 +4K. I.L.U A+T+EG+K. B+H.F+R. I. L. U A+T+E+BGK+B+H.C+8+D. F+R.I. L.U A+T+I.B.D.J+Q.L.U A+T+I+B.C+8+H.D.J+Q.L.U A+T+I.B+D.E.J+Q. L.U A-T+I+B. 0+8+H..J+Q.L.U A+T+I4.B. F+R+ +K.J+Q. L.U A+T+X+1+B. C+8+H+D, F+R+O+..J. L.U A+T.B.D+L.G.M.U A+T+B.C+8.D+L.O.M.U A+T'.B+D+L.E..M.U A+T+B+D+L. C+S.E..M.U A+T+E.B.F+R.G.M.U A+T+E+B. C+S+D+L. F+R..M. A+T.B.D+I+.HK.M.U A+T+B. C+8.D+L+G.H+K.M.U A+T.B+D~+L..H+K.M.U A+T+B+D+LG..+8.Z..HK M. U A+T+E.B. F+R.H+.M. U

-37A+T+E+B. C+8+D+L+G. F+R.H+K. M. U A+T.B+H+K.D+L. I M.U A+T+B+H+K. C+8.D+L.I.M.U A+T.B+H++D+L.B+G. I.M.U A+T+B+H+K+D+L. C+S.E+O. I M.U A+T++G. B+H+K.F+R. I.M. U A+T+E4fGB+H+K C+8+D+L F+R. I.M. U A+T+I.B.D+L.J+Q.M.U A+T+I+B. C+8+H+K.D+L.J+Q.M.U A+T+I.B+D+L.E.J+Q.M.U A+T+I+3+D+L. C+S+H+K..J+Q.M. U A+T+I+.F+R+G.J+Q.M.U A+T+I++B. C+S+H+K+D+L. F+R+G. J+Q.M. U A+T..B4L.D.G.N.U A+T+B+^.C +.D.G.N.U A+T.B4.D.E+L.G.N.U A+T+B3MDD. C+.+L. G. N.U A+T+E+L. B+.F+R. G.N.U A+T+E+L+B+M. C+S+D. F+R,..N.U A+T.B+4.D+G.H.N.U A+T+B+M.C+8.D+G.H.N.U A+T.BD+G.E+L.H.N.U A+T+B4D+G C.E+L.H.N.U A+T+E+L. B+M. F+R.H.N.U A+T+I+L+B+M. C08+D+0. F+R.H..U A+T.B 3+H.D.I+.N.U A+TM+H. C+S. D. IK K. N. U A+T.B4*H+D.E+L+. I+K.N.U A+T++H+D. C+8.Z+LG. I+K.N.U A+T+E+L+G.+M+H.F+R.I+K.N. U A +T+B+L+G +H. + +D. F+R. I+K.N. U A+T+IK.B+MC.D.. JQ.N. A+T+I+K+b+M. C++H., D. J+Q. N. U A+T+I+K. cBD.E+L.J+q.. U A+T+I+K+beD. O+8+H.E+L.J+Q..U A+T+I+K++L.B+4. F+R+.J4Q..U A+T+I+K+B+L+BL +. C+8++D. FR. J+. N. U A+T+N.B.D..O.+P.U A+T+N+B. C D. O.O+P.U A+T+N. CB+D..D.0.O+P.U A+T+N+B+D.0 C+8.E.G.O+P.U A+TI+E.3.Fi+ L.0.0+P.U A+T+N+.B.F+R+L.. O +P.U A+T++3+B. C+8+D * F+R+L. G. O+P. U A4T+N..D. H.O+P.U A+T+I+B.C4++M.D+G.H.O+P.U A+.T+N. +D+G...O+P.U A+T+N+B+D+G. CB++M..H. O+P.U A+T+N+ B B+R. +h. +L.P.U A+T+-N++3..4+S+ M+DO. F+R+L.. O+? UJ, A+T+.B+H.D.I.O+P.U A,+N+B+H.t C++M8.D.I. O+P.U A++Ni+.3B+D.4+). I.O+P.U A+T+N++H+D. C+S4. +. I. O+P.U

-38A+T+N+E+G. B+H. F+R+L. I.+P. U A+I+N+E+G+B+H. C+S+M+D. F+R+L. I. O+P.U A+T+N+I.B.D. J++K. O+P.U A+T+N+I+B. C+S+M+H.D. J+K. O+P. U A+T+N+I.B+D.E.J+Q+K.O+P.U A+T+N+I+B+D. C+S+M+H.E.J+Q+K. O+P. U A+T+N+I+E. B. F+R+L+G.J+K. O+P.U A+T+N+I+E+B. C+S+M+H+D. F+R+L+G.J++K. O+P. U

VII. CONCLUSION With the introduction of the new algorithm of a "Foldant", characteristic numbers concerning the tree are found to be obtained in a more economical and simple way than with a conventional matrix or determinant or a "primitive-node-pair connection matrix". The algorithm is algebraic as contrasted with the topological method dealing with the tree. It is so close to the fundamental idea of the tree that future investigation concerning its relationship with the other characteristic matrices of the tree derived thus far can be expected to systematize a customary matrix theory of a linear graph from a new point of view. The direct application of the algorithm is for the computation of the driving point admittance as it was discussed substantially in the thesis. If the computation of the transfer admittance by means of the foldant is as simple as the one of the driving point admittance, the algorithm would certainly compete with these prevailing methods of matrix or determinant. By incorporating Kron's method, the computation of the numerical values of the driving point admittance by the algorithm might be as efficient as by the matrix method, and the programming of the numerical computation can be developed directly from the one of the thesis. -39

VIII. REFERENCES 1. Y. H. Ku, "Resume of Maxwell's ard Kirchhoff's Rules for Network Analysis," J. of Franklin Inst., Vol. 252, pp. 211-224, No. 3, 1952. 2. W. Percival, "The Solution of Passive Electrical Networks by Means of Mathematical Trees," Proc. of I.E.E., Part III. Vol. 10, pp. 143-153, 1953. 3. F. E. Hohn, "A Matrix Method for the Design of Relay Circuits," Trans. IRE, Vol. CT-2, pp. 154-161, June, 1955. 4. M. B. Reed, "Generalized Mesh and Node Systems of Equations," Trans. IRE, Vol. CT-2, pp. 162-169, June, 1955. 5. S. Seshu, "Topological Considerations in the Design of Driving Point Functions," Trans. IRE, Vol. CT-2, pp. 356-367, Dec., 1955. 6. S. L. Synge, "The Fundamental Theorem of Electric Networks," Quat. App. Math., Vol. 9, pp. 113-126, July, 1951. 7. C. Saltzer, "The Second Fundamental Theorem of Electrical Networks," Quat.App. Math., Vol. 11, pp. 119-121, April, 1953. 8. H. M. Trent, "Note on the Enumeration and Listing of all Possible Trees in a Connected Linear Graph," Nat'l. Acad. Sci. Proc., Vol. 100, pp. 1004-1007, October, 1954. 9. I. Cederbaum, "Invariance and Mutual Relations of Electrical Network Determinants," J. Math. Phys., Vol. 35, February, 1956. 10. I. Cederbaum, "On Network Determinants," Proc. IRE, Vol. 44, p. 258, February, 1956. 11. G. Kron, "A Method of Solving Very Large Physical Systems in Easy Stages," Proc. IRE, Vol. 42, pp.680-686, April, 1954.,

UNIVERSITY OF MICHIGAN 3 9015 03483 1506