70 
Proceedings of Boyal Society of Edinburgh. [sess. 
A glance, however, suffices to convince one that the concluding 
general theorem here given differs considerably from the theorem 
which he had previously enunciated and possibly proved. As 
originally stated the theorem was — 
^ 0 a o+ 1 W i a.+ 1 ... W «n-l+ 1 j 1 
xJo +l X,Pi +1 . . . Xfa- 1 +1 
0 1 71-1 
= \ [zW 1 • • • -ifc'J 
h *°h a ' ■ ■ ■ to 1 
which being altered into the notation of his present paper by the 
substitutions 
becomes 
^0 » x i » • • 
• — x l ? X 2 J • • ' 
U 0 > u l » • • 
. = 
Po >Pi> •• 
T, Y 2 
' “ A ’ A ’ ' 
a o » a i , • • 
• = n,r 2 ,... 
A.A> • • 
. = - Si , S 2 , . . . 
A = A, 
[x^+'X/^ 1 • • • X/» +l ] 1 
,r 1 Sl+1 a , 2 S2+1 • • • x n Sn ~^ ^ 
Ty Sl Y S2 • . • Y s *i~\ 
^S!+S 2 + • • . + Sn +1 I x 1 x 2 in J 
yfyp • • • y/ n - 
Using on both sides of this the fact that if an expanded function 
be multiplied by the product of certain powers of the variables, 
any particular coefficient in the original expansion has now for 
facient its original facient multiplied by the said product, we 
obtain 
[” xfxp • • • x n Sn ~"| 
1 xp +l xp* 1 ---xjr« +l I 1 
X x X 2 • • • x n 
i r Y^Y/f-.Y,/” i 
^S!+s 2 + • • • +S w +l| ^ 1 r i +1 y 2 r2+1 • * • yn n ^ I 1 
y<yi • • • y, 
