192 
Proceedings of Royal Society of Edinburgh. [sess. 
and therefore 
there is obtained the identity 
V + Vbf • • • • A ,m, = a»*y +ad" ■ 
— 1 m 1 
(m+1) 
Then since the a’s here may denote any quantities whatever (“ qua 
in formula cum ipsa af quantitates quascunque designare possint ” ) 
a further substitution ;;; bringing us hack to differential-quotients is 
made, viz., 
f (*+i) 
a 
(i+l) _ ^ U i+\ 
*+! BX; 
k 4*1 
where u , u Y , u 2 , . . . , u m+l are functions of x x , x 2 , . . . , x m+v 
In this way 
= J 
* bx k+l 
dv, 
— UU' 
0 -m 
u 
dx,. 
+i 
and the aforesaid identity gives us 
^+2^ ± 
s'i? 
g^m+1 
du 
U 
U 
dx 1 
' 3*2 
^**’m+l 
^ ~ dx Y dx 2 
du ri 
dx, 
m+i 
as was to he proved. 
As a corollary to this it follows that if in the determinant 
y ±u . 
du x du% 
SSI dx 2 
dUn 
dxf 
tu , tu x , . . . , be put instead of u , u x , . . . , t being any function 
whatever, the effect is the same as if the determinant were simply 
multiplied by t n+ \ “ Quod iam olim alia occasione adnotavi,” the 
* A combination of the successive substitutions is impossible, by reason of 
the fact that in the second case the equation 
ji+i)_ du i+ 1 
*+! 0^+1 
is not meant, as in the first case, to include the definition of a\i + d, which has 
thus to be defined by a supplementary equation. 
The result of the first substitution is very noteworthy, in view of previous 
footnotes. 
