624 DR THOMAS MUIR ON THE GENERATING FUNCTION 



Similarly from § 13 there is obtained the theorem, 

 In the expansion of 



{m t -Tj^-cdj} -1 • {m 2 -T 2 (u 2 - w 2 )} -1 



the aggregate of the terms which are such that every r occurring in them is accom- 

 panied by the corresponding x — £ , and vice-versa, is 



m 1 - a^ix-y - f x ) - Vi^i - fi) - Vife ~ &) 



- a 2 T 2 (.T 2 - 4) ™2 _ Vg^ ~ &) - C 2 T 2( a; 2 _ 4) 



- a 3 T 3 (tf 3 - f 8 ) - Vs^s - 1 3 ) m 3 - c 3 t 3 (.« 3 - £ 8 ) 



(16) Leaving the subject of conjugal aggregates at this point, let us return for a 

 moment to the question of § 5. It will be remembered that Jacobi's first two cases of 

 a supposed general theorem were 



o 



I a l b 2 I • x x x 2 = u^a 2 - ^Mj ■ &!«! , 







I a A r 3 I ' X l X 2 X Z = U \ U 2 U S ~ 2/ M l ' (V'l ;i l + \ C 1 X 2 + \ C Z X i) * X l > 



and that for three different reasons it was difficult to avoid a strong presumption that 



the form of the general theorem must be 





 I ajb 2 c z ... I • x x x 2 . . . x n = ttyU.y . . . u n - 2,2*! ' -^1 ' x i 



where F x is an integral homogeneous function of the (n — 2) th degree in x\ , x 2 , . . . x n ; 

 A fourth reason will now incidentally appear, it being inseparable from certain curious 

 results in determinants which were discovered in consequence of it, and which deserve 

 on their own account to be chronicled. 



Early in the investigation it was observed that the expression following 2/u x in the 

 second case above is equal to the determinant 



U 2 ~~ C '2 X 2 ~ C Z X Z 



- l 2 x 2 u 2 - b 3 x z 



and on turning to the first case it was seen that the corresponding expression there, viz.,. 

 b x x ly being equal to u 2 — b. 2 x 9 , could be considered to be of the same form. The 

 question thus arose whether or not there might be a similar determinant of the third 

 order having (1) x 1 for a factor, (2) a quadratic function of x 1 ,x 2 ,x 3 ,x i for the 

 cofactor, (3) the terms of this function sixteen in number and all positive, and (4) 

 these terms such as to ensure an identity ; and, strange to say, determinants possessing 

 the first three of these properties were found to be realisable. 



(17) If u x = a x Xj + . . . +a 4 x 4 , u 2 = bjXj + . . . b 4 x 4 , . . . . each of the six deter- 

 minants. 



