1903 - 4 .] 
Dr Muir on General Determinants. 
67 
The coefficient of — in the expansion of — - — , 
; xT •'(a 0 x + b 0 y)“+ 1 (b 1 y + a 1 x)“+i 
is the same as the coefficient of t 0 m t 1 n in the expansion of 
(bitp bp^i)^ 1 ( a pt 1 a ifcp) v 1 
I apbi h *"- 1 
, it being remembered that m and 
n are of the same sign as p and v respectively and that m + n = 
p + v - 2 . 
In similar fashion the author deals with the case of three 
functions u 0 , u Y , u 2 of three variables x , y , z , proving labori- 
ously and not very neatly the neat result 
I °o \ 1 • (-rjr 
/ 1 
1 \ 
/ 1 
1 \ 
/ i 
1 \ 
1 
+ 1 
+ : 
<f — xj 
\y-r) 
y-y' 
z-*/ 
thence deriving 
t 0 n V V - V 
m+1 uf +1 u 2 r+1 x^ +1 y v+l zp +1 
and ending with the theorem : — 
The coefficient of - - ------ in the expansion of 
1 
(a 0 » + b 0 y + c^y^ib-pj + cp + a 1 x) n+1 {c 2 z + a 2 x + b 2 y) r+l 
is the same as the coefficient of t 0 m t 1 n t 2 r in the expansion of 
{ l&iC#o + l& 2 c ol^ + W c \\hY ~ 4 \ c vNh + \c 0 a x \k + l c i ft 2 ^o} v ~' 1 { l«p6il<a + l«i&a^o + l«2&o|*i } p ~ 1 
\af x c^+v+p-‘i 
it being understood that m, n, r are of the same sign as p, v, p 
respectively omd that m + n + r = //, + i/ + p- 3. 
The corresponding results for n functions of n variables are 
evident. They had already been enunciated in the introductory 
section of the paper, and Jacobi now merely adds “ Omnino 
similia theoremata de numero quolibet variabilium, quae § 1 
proposuimus, eruuntur.” It has to be noted, however, that belief 
