198 



DR THOMAS MUTR ON 



where the elements which may be all different are 



the [n 2 , which here is] 4 2 elements of | a 1 b 2 c 3 d i | , 



the [n—l, which here is] 3 elements a 5 , /3 5 , <y 5 , 



and the [^n(n — 1), which here is] 6 elements D 12 , D 13 , D 14 , 



^23> ^24 ' 



'34 



This degenerates into the identity of the preceding paragraph if we put D 12 , D 13 , D 14 



= & 2 ' h » bi ; D 23 = ^4 5 ft = ^i • 



(16) Having thus the general fourth-order theorem of which (N 4 ) is a case, it is 

 natural to inquire whether there be a corresponding general fourth-order theorem of 

 which (D 4 ) is a case. The following new result regarding the sum of two determinants 

 of the fourth order gives the answer to this inquiry : — 



If | a 1 b 2 c 3 d 4 | , | a 1 /3 2 y 3 ^ 4 | be any two fourth-order determinants, their sum is 

 equal to the sum of six determinants, each of which contains a pair of complementary 

 minors from each of the originals, viz., the sum 



+ 



a i a 2 



73 



74 





a x 



a 2 



ft 



ft 





a x 



a 2 



«3 



a 4 



b x b 2 



a X «2 



^3 

 C 3 



<*4 

 C 4 



+ 



«1 

 c l 



a 2 



C 2 



b 3 

 «5 3 



b 4 



^4 



+ 



ft 



7i 



ft 



72 



b 3 



C 3 



b 4 



C 4 



ft ft 



d 3 



d, 





7i 



72 



d 3 



d 4 





dx 



d 2 



K 



<?4 



«X «2 



a 3 



a 4 





ft 



ft 



S 3 



a 4 





7x 



72 



a 3 



a 4 



\ b 2 



C X C 2 



ft 

 73 



ft 

 74 



+ 



«x 



b 2 



<*2 



«3 



C 3 



«4 



C 4 



+ 



*x 

 c x 



<?2 

 C 2 



b 3 



«3 



b 4 

 a 4 



s, s 2 



d 3 



^ 





dx 



d 2 



73 



74 





dx 



d 2 



ft 



ft 



Expressing each of the six as an aggregate of products of complementary minors, we 

 have the sum equal to an aggregate of thirty-six products, viz. : 





«X «2 



C 3 C 4 



- 



&•* din 

 a X a 2 



&, 5 4 



d 3 d 4 



+ 



a x a 2 

 ft ft 



&$ ^4 



C 3 C 4 



+ 



«X «2 



73 74 



- 



ft ft 



73 74 



c 3 c 4 



+ 



«X a 2 



Iftftl 



y 3 y4 

 Is, si 



+ 



«x « 2 | 



a l «2l 



S 3 <$ 4 



d s d i 



a x a 2 

 i c 1 c 2 





+ 



a x a 2 

 7x7 2 



& 3 & 4 



S 3 Si 



+ 



«X a 2 

 C X C 2' 



ft ft 



d 3 d t 



- 



«x « 2 

 7x72 



ft ft 



<5 3 5 4 



+ 



C l C 2| 



7x7 2 l 



6 3 &J 



«x« 2 



ft ft 



6 :i C 4 



5 3 Si 



- 



7x7 2 



s 3 s 4 



+ 



a x a 2 



c 3 c 4 



+ 



ft ft 



7x72 



a 3 a 4 



— 



ft ft 

 d x d 2 



«3 a 4 



c 3 c 4 



+ 



7x7 2 

 d x d 2 



«3 a 4 

 ^3* 4 



+ 



«x <* 2 



73 74 



rf 3 d 4 



— 



a x a 2 



C X C 2 



ft ft 



^3 d i 



+ 



«X «2 



<?X <*2 



ft ft 



73 74 



+ 



c x c 2 \ 



a 3 a 4 

 rf 3 rf 4 



— 



S x S 2 



a 3 a 4 



73 74 



+ 



c x c 2 



o 3 a 4 



ft A 



+ 



ft ft 

 »1 h 



C 3 C 4 



7(74 



— 



ft ft> k «4 



S 1 S 2 \\ y 3 y 4 



+ 



ftft| 



<^ d 2 \ 



a 3 a 4 



c 3 c 4 



+ 



$1 ^2 



a 3 a 4 



73 74 



— . 



d t d 2 



c 3 c 4 



+ 



S x S 2 

 d x d 2 



a 3 a, 

 a 3 a 4 



+ 



7i7 2 



^1 ^2 



! Og «4 



ft ft 



- 



7i7> 



C X C 2 



ft ft 



+ 



7x72 

 rfj c£ 2 



& 8 *4 



a 3 a 4 



+ 



s x s 2 



h C 2 



a 3 a 4 



ft ft 



- 



d x d 2 \ 



a 3 a 4 

 a 3 a 4 



+ 



d x d 2 \ 



j a3 a 4 

 1 63 J, . 



