the calculus of functions. 
4°5 
then we find 
which fulfils the given conditions. 
In the same manner it may be found, that the function 
f i — ax 1 + x* 7 
^ 1 i -J- ax 2 + x* j 
will remain the same, whether the variable is x, — x, or — . 
Problem VII. 
Given any two series of functions a x, (3 x, &c. v x and ax, 
i 
x , &c. vx, required the form of p, so that the following equa- 
X I 
tions may be fulfilled, 
p a X =■ p a X 
i 
p (3 X = p ft X 
I 
&c. &c. 
p v X = p V X. 
i 
Determine/ from the equation 
f a x —f aX ( a ) 
i 
take some particular case, and determine/ from the equation 
i 
ffcc (3 X =ffccp> X = ffa /3 X (b ) 
i iii ii 
take a particular case and find/ from the equation 
2 
fffa ( 3 yX =fffu ( 3 yX =fffafi y a? (c) 
2 1 
2 1 
1 1 
2 i 
and continue this to 
ff. .faf 3 y . .vx =// . .fap>y..v X ( n ) 
