190 Proceedings of Royal Society of Edinburgh. [sess. 
Leaving now these general theorems which involve two suffixes 
m and n , and which concern groups of functional determinants, 
Jacobi returns (§ 17) to the consideration of the properties of a 
single functional determinant, the specialisation being made, not 
by giving a particular value to m the second suffix introduced, 
but by leaving it unrestricted, and putting the original n — 1. In 
the theorem 
a) 
2 ± M . 
B then stands for 
± ¥ 
dx dx 1 
djf 
dx n 
ra+l 
b {m = B" 
m 
Y. and if for "V +— . . Making in 
dx k ^ ~ dx dx k+1 
this the further specialisation, /= 0, so that x has to be considered 
as a function oi x 1 , x 2 , ... , X m _ x we have 
+ 
*+i 
df dx 
dx dx ; 
&+i 
and consequently 
i.e. 
<f_ ' dfi+i 
dx dxj 
&+i 
fc+i 
¥ (V< 
= 
dx \ 
■k+1 
¥_ 
dx 
i+l _j_ i+l 
dx*,-, dx 
ty t+i 
dx 
dx ) 
dx*. i_i f 
fty i+l \ 
\dx k+1 ) 
if the brackets in the last line be taken to indicate that the f i+l 
enclosed by them does not involve x but its equivalent in terms of 
x 1 , x 2 , ... , x m . By use of this substitute for there results 
*i.v + ^Y 0 AY. . . . I v 
)x ~ \dx 1 )\dx 2 ) Wm+i) " 
+ <f m ¥i 3/m+l 
dx dx. 
dx „ 
df 
and if we denote the cofactor of — in the determinant on the right 
dx k 
by A k we have of course the said determinant 
.0/ ■ df 
= A d~x + A 'te x + ' 
^ A ^ _ A — — 
dx 'dx ‘ dx. 
¥ 
UJj m + 1 
df dx 
“ +1 a*- dx m+1 ’ 
and .•. 
