180 Proceedings of Royal Society of Edinburgh. [sess. 
the equivalents here given for 
df df_ df | 
, dx ’ dxf ' ’ ‘ ’ dx n ’ 
there is obtained a determinant which is expressible as the sum of 
n+ 1 determinants, all of which vanish except the first. We thus 
arrive at 
# ; <fn = ¥1 ■ ¥n 
^ ~ dx dx x dx n ^ — dx dx x dx n ’ 
as was to be proved. 
Passing to the case where two of the functions are changed, he 
says, first, that if by the use of the equations 
</> = a > fz = a 2 > fd = a 3 » • • • > fn = a n 
the function f is changed into <f>\, then exactly as before it can be 
shown that 
y + d -i . d A . . . . A = y + d l . d h . Ms A 
^ ~ dx dx x dx n ^ ~ dx dx x dx 2 dx n ' 
More questionable is the logic of his second step, which is to the 
effect that from this and the previous result it follows that 
y + A §/» — y + A df df n 
^ ~ dx dx x dx n 1 — dx dx x dx 2 dx n 
His third step is simply the assertion that by proceeding in this 
way we may prove generally that if by use of the equations 
f= a , /i = a i5 • • • • = + a 
fi becomes changed into then 
■y _j_ df df cf n __ ^ ^ dcj> dcf>i d<j> n . 
^ “ dx dx x dx n J “ dx dx 1 dx n ’ 
and the matter is concluded with the further assertion that if in 
the elements of the second determinant there be substituted for 
a , cq , . . . , a n the functions which they represent, that determin- 
ant will be identically equal to the other. 
Considerable space is next given (§ 15) to the discussion of the 
case where the number of variables, x , x x , ... , x n+m which the 
