212 
Mr, Babbage's essay towards 
Problem XVI. 
Given the equation 
F {4' {x,y), • • &C. -i/ p ’p(x,y) 1=0 
The same substitution as that employed in the last Problem 
will reduce this equation of two variables to a similar one 
which contains only one 
putting^ we have 
«• (X, V) 
v X.'n- t-i 
and {x,y) = <p ) also making ~ = v 
the given equation becomes 
— — n 
F | c pv, <p z u, . . <p?U | =0 
an equation which contains only one variable, and may there- 
fore be solved by the methods described in the first part. 
Problem XVII. 
Required the solution of the equation 
F {'4' ( x >y)> ¥’ 2 (x>y)> • • ty p,p ( x >y)} = o 
The following considerations lead to another mode of solu- 
tion applicable to this Problem. If in the function $(x t y) 
we put y equal to x it becomes $ (x, oe) call this cpx : then if 
in the second simultaneous function of i]/ (x,y) we put y equal X, 
the result will be the same as if we had taken the second 
function of ip (x, x) or <p x relative to x, or symbolically ex- 
pressed, it is 
V' \ x >y) = ^ (*> x )> i x > x ))> = Ly=x] 
this may be rendered evident by substituting for the right 
side of the equation its value (x, y), (x,y)). 
