250 
Mr. Babbage’s essay 
towards 
and 
d *4/ («a?» y) 
dxdy 
d*4/ (x, %) 
dx 7. 
hence 
d l fy (x, @y) _ 
da.x d. ( (#> @y)[d(2y 
dx 1 
dx dy l 
dy t dy 
put (3y for y, observing that (||) = { | (j|] ] = ^ 
and also , then there will result the 
equation 
dfiy d z 4 / ( x , y ) dxx d. z 4 ' (x, y) 
dy dx % ~~~ dx dy x 
This is a partial differential equation from whose solution 
$ (x, y) may be found. 
Ex. l. Given the equation —^ x ’ ^ in this 
case ax=a — x and (3y = b — y t and the differential equation 
to be solved is 
d % 4 / (x, y) __ d 7 - 4 / ( X , y) 
dx 1 dy * 
and its solution is 
■H x >y')~ < p( x +y') + ( p{ x —y) 
the two arbitrary functions cp and <p must be determined so as to 
fulfil the given equation, for which purpose we have 
— x+y) — <p {a — x — y ) 
and^^ ) =Kb + a:-^)+<p(-6 + a: + ^) 
q> and <p being respectively the differential coefficients of <p and 
2 
<p, since these two expressions must be equal, we have 
<P (a — x —y) = i p{b+ x—y) 
■ — <f> (a — x + ;y) = <p ( — b + x + J0 
and 
