622 



DR THOMAS MUIR ON THE GENERATING FUNCTION 



terms in the expansion of the reciprocal of t^ ' t 2 u 2 ' r 3 u s is the reciprocal of 



i.e., T(t iUi ■ t 2 u. 2 ■ T3M3)- 1 = I aj) 2 c 3 h 1 • (t^j • t 2 x 2 • t 3 x 3 )~ 1 . 



The similar variant of the other theorem of § 9 is 



+ + 



T(t 1 W 1 • T 2 U 2 • T3W3) = I OjlVs I • ( T i a? l • T 2*2 • Vs) ' 



(11) Knowing the truth of this second theorem in its general form, viz. : 



+ + 



T(t a m a • r k u k ■ Tfa ■ • ■ ■ ) = I hjfah • • • I • (T h x h ■ r k x k ■ T t X; • • • ) 



we can readily find the aggregate of the conjugal terms in any series 



Ci</>! + G 2 <t> 2 + C 3 <f> 3 + ■ ■ • • 

 where the <p's are of the form r h u h ' r k u k ' T t u t ' ' ' ' . For example : 



"YiGT^Uy ' T 2 U 2 + Dt 2 U 2 ■ T S U 3 + ET3W3 • T-jUj 



+ + + + + + 



= C I a-fi.2 I • t 1 x 1 . t 2 x 2 + D I b 2 c 3 I . t 2 x 2 • r 3 x 3 + E | c 3 a x \ ■ t 3 x 3 • r 1 x 1 . 



(12) A conspicuous instance of such a series is the expansion of (wij-^Wj) 

 (m 2 — r 2 u 2 ) (m 3 — t 3 w 3 ) . . . . Eestricting the number of factors to three, merely for 

 simplicity's sake in writing, we have the aggregate of the conjugal terms of this 

 product 



= Ti^hPVH - ( m i m 2 • T 3 M 3 + ■■••) + ( m i • T 2 U 2 ■ T 3 W 3 + ) " T 1 M 1 " T 2 M 2 '^sl 



+ + + + 



= m 1 m 2 m 3 - (m 1 m 2 ■ c 3 t 3 x 3 + • • • ) + (m± • | b 2 c 3 | • r 2 x 2 ■ t 3 x 3 + •■••)-) a x b 2 c 3 \ t^ • t 2 x 2 • r 3 x 3 



+ + 



m \ ~ a \ ' T i x i — ^1 ' T r c i — c i ' T i a 'i 



- a.-, ■ t 2 x 2 m 2 — b 2 • r. 2 x 2 — c 2 • r 2 x 2 



— a 3 ■ T3X3 — » 3 • T3.X3 to 3 — r.'g t 3 x 3 



the reason for the last step being the fact that the law for the expression of a 

 determinant with binomial elements in terms of determinants with monomial elements 

 applies to the case of permanents also. 



(13) This theorem has corresponding to it a per-reciprocal theorem, viz., 



~X{{m l -r l u l )(m 2 -T 2 u 2 ){m 3 -T 3 u 3 )} 1 = 



m i ~ a i T i* c i 



b, ■ T,X, 



e, • t-,x. 



- a 2 • t 2 x 2 m 2 — b 2 - r 2 x 2 — c 2 • r 2 x Si 



"3 ' T 3 X S 



2 2 

 m 3 ~ C 3 ' T 3 X 3 



( T 1 X 1 ' T 2 3 '2 * ^s)" 1 



By way of proof let it be noted that the determinant, A say, whose reciprocal appears 

 on the right may be written 



(j«j - TjMj) 4- t^Wj - a x x^) 



— Tc, ■ Ci.yX.-, 



~ ~ To ' iX&JUn 



— T 1 • ftjiCj 



(m 2 -T. 2 u 2 ) + T 2 {u 2 -b 2 x 2 ) 



T 2 ' C 2 X 2 



~ T 3 • 6 3 X 3 ( m 3 ~ T 3 M s) + T s( M 3 ~ <V e ») 



and that this when expanded becomes 



