Mr. Babbage’s essay towards 
Let y = ip x be the equation of the curve, also AN = x, 
and if AN x AM = a*, then the property of the curve is that 
PN = QM, 
but PN — y — tyx, and OM = ^ (AM) = 4' |~j conse- 
quently the equation from which vj/ must be determined, is 
a* 
■^00 = 4 / - 
its solutios in 
if we makey = x it becomes yx — x* = a ? , an equa- 
tion to the hyperbola. 
Problem III. 
Required the general solution of = Ax 
having a particular solution, and also one of -^x = a 2 x. 
Assume \p x =fx x cp [fx,fv J 
I 2 
making this substitution in the original equation, it becomes 
fx x (p \fx>foc | == A x x fa x cp \fuxjax ] make/ a x —fx, and 
fax —fx, from this it results that fx —far x of which we pos- 
I 2 II 
sess a particular solution, divide both sides by <p\fx,fax\ then 
i i -> 
we have 
fx = Ax x/ ax ; 
this is nothing more than the original equation of which, and 
of fx — fa," x, w r e have by hypothesis particular solutions : 
