214 



DR THOMAS MUIR ON THE 



and 



( a h g )-» = ( A H G ) 



A b f D D D 



g f e H B F 



D D D 



G F C 



D D D 



( a h + v q — n ) _1 ( cof (a) cof (A — y) coi(g + /x) ) 



A A 



h-v b f+X 

 g+H f-X c 



A 

 cof (A + v) 



cof (5) cof(/-X) 



AAA 

 cot (ff -/jl) cof(/+X ) cof(c) 



AAA 



Consequently the product of the four matrices on the left side of the identity is ;i 

 matrix whose (1,1) element is # 



A 

 D 



H 

 D 



G 

 D 



a 



A 



9 



a 



h + v 



9-u. 



cof (a) 



A 



cof (A — v) 



A 



cofto+M) 



A 



h — v 



b 



/+x 



cof (A + j/) 

 A 



cof (6) 



A 



cof(/-X) 

 A 



9+1* 



/-x 



c 



cof (0-/*) 



A 



cof(/+X) 



A 



cof (c) 

 A 



and the product of the two matrices on the right is a matrix whose (1,1) element is 



cof (a) 



cof (A — i/) 



coi(g+fx) 



A 



a , h+v 



or, as Cay ley would have written it, 



</-M 



^cofjo) ) cof(A- y)> cofto+g)^ h+v> y.^y 



We have thus to prove the identity of these two elements, and eight other identities 

 like it. 



Multiplying both sides by DA we see that this is the same as to prove that 



H 



G 



a h + v g — fA 

 h-v b f+X 

 // + M /-X c 



h 



cof (a) cof (A — v) coi(g + fjL) 

 coi(h+ v ) b cof(/-X) 



cof to-/*) cof(/+X) c 



= D 



cof(a), cof(A — v), col(g + fi) 



h + > 



g-u 



But the left-hand member of this can be partitioned into 



A 



H 



G 



a 



h 



9 



h 



b 



./' 



!/ 



f 



c 



h 



cof (a) 



+ 



H G 



V -fi 



— V . X 



n —X . 



* Trans. Roy. Soc. Edin., xxxii. pp. 461-482 



cof (a) 



