618 DR THOMAS MUIR ON THE GENERATING FUNCTION 



give at once for l n , 2 22 , 3 33 , 4 44 the values 



o^c^a-y , c 2 d 2 a 2 > d 3 a 3 o 3 , a i b i c i . 



Taking next the coefficient of x 1 ^x 2 , we have 



a il-i2 + a 2 l u + & X 2 U = a 1 b 1 c 1 d 2 + a 1 b 1 c 2 d 1 + a 1 b 2 c 1 d 1 + a.-p^dy , 



which, in consequence of the previous determination of l n , is reduced by a term on 

 both sides. Consideration, therefore, of this coefficient and the eleven others like it 

 will be seen to lead to the twelve equations 



a x l 12 + 6 1 2 11 = a^Cjrfj + a^b^dy + a 1 ^ 2 c 1 rf 1 , (2) 1 



6 2 2 23 + C 2 3 22 = b 2 C 2 d 2 a 3 + h 2 C 2 d S, a -2 + <Va tf 2 a 2 » ( 2 ) 2 



c 3 3 34 + d 3 4 S3 = c 3 d 3 a 3 b i + c 3 d 3 ap 3 + c 3 d 4 a 3 b 3 , (2) 3 



d 4 4 14 + a 4 l 44 = «i 4 a 4 6 4 c 1 + d i a i b 1 c i + d^a-J)^ , (2) 4 



a 1 l 13 + c 1 3 n = a 1 b 1 c 1 d 3 + a-fi^dy + a 1 6 3 c 1 c? 1 , (3) 1 



6 2 2 24 + <2 2 4 22 = 5 2 e 2 d>a 4 + b. 2 c 2 d A a 2 + b 2 c 4 d 2 a 2 , (3) 2 

 c 3 3 13 + a 3 l 33 = c 3 d 3 a 3 b x + c^a^ + c^a^ , (3) 3 



^4 4 24 + & 4 2 44 = <^ 4 «A C 2 + rf W ? 2 C 4 + d i a 2 b i C i> ( 3 ) 4 



a 1 l li + ^4 n = a^Cyd^ + a-fi^c^ + ap 4 c x d x (4) x 



Z> 2 2 12 + a 2 l 22 = b 2 c 2 d 2 a x + b 2 c 2 d x a 2 + b 2 c x d 2 a 2 (4) 2 



C 3 3 23 + ^3 2 33 = C 3^3 a 3 & 2 + C Z d 'i a 2 b 2 + C 3^2 a 3 & 3 ( 4 ) 3 



' ? 4 4 34 + C 4 3 44 = ^ 4 a A C 3 + ^4«4 5 3 C 4 + d 4 a 3 h i C i (*) 4 



Dealing next with the coefficients of x 1 2 x 2 2 , x-^x^ .... we obtain 



a x \ 22 + a 2 \ n + b 2 2 u + \2 12 = a 1 b 1 c. 2 d, 2 + a 1 c l b 2 d 2 + a x dp 2 <: 2 + b^a^ + b^a.f^ + c^d x a 2 b 2 (5)j 



6 2 2 33 + & 3 2 23 + C 3 3 22 + C 2 3 23 = b 2 C 2 d -iH + Ws^S + b 2 a fz d 3 + C 2 t? 2 6 3 a 3 + C 2 a 2 b 3, d i + ^2 a 2 & 3 C 3 ( 5 )2 



C 3 3 44 + C 4 3 34 + ^4 4 33 + rf 3 4 34 = H d i a i b i + C 3 a 3^4 & 4 + H\ d i a i + d S a 3 C Jh + ^3 & 3 C 4 a 4 + «3 6 3 C 4^4 ( 5 )3 



d 4 4 u + c? x 4 14 + Ujl^ + o 4 l 14 = c^a^Ci + ^ 4 6 4 a 1 c 1 + d^a-p x + ajb^d^ + a^d^ + & 4 c 4 ^ 1 a 1 (5) 4 



ajlgg + a 3 l ]3 + c 3 3 n + c t 3 13 = afi^ds + a-^b^ + ^VV's + 6 i c i a 3 rf 3 + Vi a 3 c 3 + c^a^ (6\ 



6 2 2 44 + 6 4 2 24 + <2 4 4 22 + r? 2 4 24 = b 2 c 2 d i a i + b 2 d 2 c 4 a 4 + b 2 a 2 c±d± + r^dp^ + c 2 a 2 6 4 eZ 4 + <2 2 « 2 & 4 c 4 (6) 2 



These six, it will be found, do not contain any other unknowns than those contained 

 in the previous set of twelve ; and, what is still more important, the unknowns con- 

 tained in any one of the set of six are exactly those contained in two of the set of 

 twelve. For the purposes of solution, therefore, it is better to view the eighteen as 

 belonging to six sets of three, viz., 



(5)! , (2): , (4), 



(5) 2 , (2) 2 , (4) 3 , (6) x , (3) 4 , (3) ( 



and 



/3 



(5) 3 . (2) 3 , (4) 4 (6) 2! ( 3 ) 2 ,( 3 ) 4 - 



(5) 4 > ( 2 )4 . (*)i 

 Taking the first triad, viz., 



a 1 l 22 + a 2 l 12 + 6 2 2 11 + & 1 2 12 = a- i b 1 c 2 do + . ■ - 



« 1 1 12 + /; 1 2 11 = ap^c^ + ap x c 2 d x + a-fi^d-y 



a 2 l 22 +^2 2 i2 = b 2 c 2 d 2 a x + b 2 c 2 d x a 2 + b<f x d 2 a 2 



we see that there are four possible solutions of the third equation ; and, as substitution 



