616 DR THOMAS MUIR ON THE GENERATING FUNCTION 



theorem, but virtually involves the generalisation, and makes the enunciation of the 



latter a mere matter of form. Let us see, however, whether it be so in this particular 



instance. 



In the first case Jacobi. abruptly inserting j a x b 2 | into the numerator of the given 



fraction, says that 



a x b 2 - a 2 b x a, 1 b x 1 



(a x x x + a 2 x 2 )(b 2 x 2 + b x x x ) x 2 a x x x + a 2 x 2 x x b 2 x 2 + b x x x ' 

 also that 



rtj 1 1 1 a 2 



and that therefore 



a x b 2 - a 2 b y 1 1 a 2 1 b x 



(a x x x + a 2 x 2 )(b 2 x 2 + b x x x ) x x x 2 x x a x x x + a 2 x 2 x 2 b 2 x 2 + b x x x ' w 



a glance being then all that is necessary to make clear that the expansion of the last 

 two fractions on the right consists of terms involving negative powers of one variable 

 and positive powers of the other, and consequently that the cofactor of x{ 1 x 2 1 in 

 (a x x x 4- a 2 x 2 )~ l (b 2 x 2 + byX-y)' 1 is (a x b 2 — a 2 b x )' 1 . 



In the second case, after certain preliminaries regarding subsidiary functions which 

 are necessary for the expression of his final result but which have nothing to do with 

 the theorem at present before us, he with equal abruptness throws down the " easily 

 verified " identity 



I a i^2 c 3 I ' x i x i x z — UiU 2 n 3 - k 1 m 1 (6 1 c 1 9: 1 + c x b 2 x 2 + b x c 3 x 3 ) 



— x 2 u 2 (t: 2 a 2 x 2 + a 2 c 3 x 3 + c 2 a x x x ) 



- x s u z( a 3 h s x % + \ a \ x \ + a z h -P'-i) ' 



and dividing by x x x 2 x s ' u x u z u z obtains of course 



j Cl-Jjfa j 1 byCyXy + Cybflt^ + ^jCjCCg 



(<»!*!+ . . • )(\X 2 + . . . ){C Z X 3 + . . .) " X X X 2 X 3 X 2 X z (b 2 X + . . . )(C 3 « 3 + . . . ~) 



2 2 2 "^ CirtC-oXo ~r Cn(Jl/-tJC-t 



(b} 



s 1 \ *t ^ "" ' ' ' /\ 1 1 "^ * ■ • / 



a 3 b 3 x 3 + b 3 a x x x + a 3 b 2 x 2 



a?,a: 2 (a 1 !B 1 + . . . ){b 2 x 2 + . . . ) 



This is the analogue to (a) in the previous case, and reasoning with it exactly as 

 with (a) he concludes that the cofactor of x x x x 2 x x 3 l in (a 1 x 1 + a 2 x 2 + a^x 3 )~ l 

 (b 2 x 2 + . . . )-\c s Xz+ . . . )- 1 is | a L b 2 c s j" 1 . 



(4) The natural commentary upon this is that there is a certain arbitrariness in the 



mode of procedure in both cases : that there is no similarity until the final identity is 



reached : and that the result of this similarity is to force upon us the conviction that 



for the establishment of the next case of the theorem there must be an identity (c) 



similar to (6) and (a). Now an examination of the last two fractions in (a) and the 



last three fractions in (6) shows that each set forms a cycle, the generating substitutions 



in (a) being 



1,2 



[,b,a ) , ^2, 1 ); 



