624 



DK THOMAS MtjIR ON 



(3) The next case, viz., where the given determinant is of the 6th order, admits of 

 two forms of development, the compound determinants in the one being of the 3rd 

 order with elements of the 2nd order, and in the other of the 2nd order with elements 

 of the 3rd order. 



Taking the latter first, we have, as before 



- l«A/sl 

 + I <h c J% I 

 + I <\ c Ji 



- I \hA 



- I KhA 

 + I ¥2/3 1 



^5/0 



Ma 

 \ c A 



a A d r e 6 



a 4 C 5^G 



ajb 5 d 6 



+ 

 + 



+ 



I a r d.,c 6 I 



I ^l C 2 f ^3 ' 



! \d 2 e s I 

 I hdtfz i 



I We I + 



I V 5 / | - 



I We I " 



a A C Jt\ + 



aj> 2 e s I 

 ajC. 2 e 3 I 



a A?3 



V-' g 3 I 



M2/3I 

 ^3/3 I 



Me/el 



a 4 c 5 e fi I 



and as the sum of the first and last terms of this development is equal to 



a A C 3 



d\ e -iJz 



a 4 6 5 c 6 

 «We 



the sum of the second from the beginning and the second from the end equal to 



I « 4 Mg I j 



i C 4 6 5./« I I ' 



W» 



and so on, there results the ;identity 



1 \ a &A d AA 



! £ «M I ^s/e 



there being ten terms on the right included under the sign of summation, and the sign 

 preceding each being the same as the sign of that particular term of the original deter- 

 minant which is brought into prominence by the notation employed. 



Taking next the expansion of | iHjb. i c s d i e 6 f e | in] terms of minors of the 2nd order, 

 we should obtain 90 terms of the form I 



I a^ I • I c./ 4 I • I e 6 f 6 I ; 



and these we should find capable of being collected into 15 sets of 6, with each set 

 expressible as a permanent of the 3rd order, the identity reached being 



hf>^Ahf 6 1 = 2, 



(4) Towards the establishment of the theorem in all its generality the first step 

 necessary is to prove the following : — 



Tf the ro>cs of a determinant of the (inn)" order be separated by horizont<d lines into 

 n sets ofm rows each, and the columns l>e similarly divided, the result may be viewed 



a A I 



! «A 1 



1 «5 & (i 1 



c x d., | 



1 c z d t | 



1 c A 1 



e iA\ 



lfi/*l 



1 hU 1 



