496 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
§ 4. The Theory of Thi'ee Identities. 
24. The course of the investigation at this point necessitates the introduction of 
two identities similar to, and in addition to, the fundamental identity. 
Let 
1 a^^fc + a^yj +... + + ... = ( 1 + (1 + + ^zV) .(I-) 
1 + hi^x + + .. , + hjj,^x'^f'^ -j- (1 + + /S.ff). . ^ (II.) 
1 d- CiqX + CQiy +... + Cptfc^'y'i +...= (!-}- ctf'^x + ^f^y) (1 + a.ftx + (iff )... (III.) 
wherein x and y may be regarded as undetermined quantities and the identities as 
merely expressing the relations between the coefficients on the left and quantities 
a, /S on the right. 
Assume the coefficients and quantities in the first two identities to be given and 
the coefficients in the third identity to be then determined by the relations :— 
1 + Cqi^ + + . . • + + . . . 
= f^s (1 + + • • • + + •••)> 
^ and y being undetermined quantities. 
Multiplying out the right-hand side of this relation, it is found to be equivalent to 
the series of relations ;— 
^10 — ( If *) *^10 > 
^‘oi ~ ^U1 5 
cao = (20)6,o + (ro') 
ffii = (ll)6n + (Td bl) 
Co, = (02) ho, + (bl^) hoP , 
ffio = (30) ^30 + (2b Tb) h,o?qo + (Tffi) , 
c,i = (21) h,^ + (2b bT) h,o5oi + (11 rb) ^ihjo + (10^ bl) hjo'hoi , 
Cj, = (l2) h|3 + (02 10) ho,h^o "b (H 01) hj^^ho^ + (fbOffi) , 
= (03') ho3 + (b2 OT) ho,hoi + (bffi) hoi^ ; 
