174 
Proceedings of Royal Society of Edinburgh. 
y + d l .Ai . . A . y + ¥ .A . ... . 8 Z». 
“ 0/ ' df Y df n dx ' dx 1 dx n . 
It thus follows that the result sought is 
y 0F 0F X 0F W 
y + WV 1 A = (_ 2 \^+ 1^™^ 0fl! 0*U^ 03?},, 
dx n ~ y ~ ; + 0F aFj 0F„ ’ 
^ J± 3/ s/i 3/» 
a theorem which Jacobi again takes pains to have noted as the 
analogue of the theorem which holds when F (/, ic) = 0, viz., 
df _ _3F_^0F 
dx 0 / * dx 
By way of corollary it is remarked that as the equations 
F = 0 , F 1 = 0 , , F„ = 0 
cannot be more appropriately viewed as giving the f’s in terms of 
the a?’s than as giving the a;’s in terms of the f s, we therefore have 
the twin result 
v + d l . • ••• • d A 
V + Z Z] . . ■ ' CX " = I _ 1 V«+l . ~ W 3/l <fn 
^-dfdf 1 df„ K ’ + 0F 3Fj 0F„’ 
~ dx dx l dx n 
and consequently from the two a theorem already obtained by a 
different method, viz., 
f dx dXj dx n 1 
^ ' ' ’ ' ¥ = y df_ \ ¥ ¥’ 
~ dx dx Y dx n 
Steadily pursuing his analogy, Jacobi next takes up (§ 11) the 
•case where the /’ s are not given immediately in terms of the x’s, 
hut are given in terms of functions </> , <f> 1 , . . . , <]> p of the x’s. 
Here, of course, we have 
dfi 0/i dcf> df i 0</> 1 df i dcf> p 
dx k ~ 0c p * dx k + d^dx k + * * * + d<t> p * dx k 
and therefore by (n + l) 2 substitutions there is obtained for 
^ + df_ df, dU 
2j ~dx dx, dx n 
an equivalent determinant, each of whose elements is the sum of 
p + \ products. This latter determinant, however, we know from 
