616 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION 



d 



'- — - (h (z\ = a, o.n 

 d 



— J- (s) = a const. 



= A suppose ; 



d-' 



and 4> (z) — -5—7- A 



' ^ dz^ 



or 



d s _ A 

 dx \/« 



the differential equation to the cycloid. 



Lest any difficulty should be felt in this example from the value of A being 

 apparently zero, we think it advisable to write down the full value, which we can 

 easily do by retracing our steps : 



^= -^ (-1)^ /— _sin07r . _^ ^M 



since m— —^ p = -^•. 



hence, if a be the constant time, we get 



.,. \/2g -1. rfi 



^TV—l sinOTT dz^ 



_ V2(/ —1 sinOvr s a 



_ \/'2(/ a 

 ds _ f^'2 a a 



Hence -^ — • -^ a result which coincides with that obtained by the 



ordinary process of expansion. 



The facihty which this process affords in the solution of the more simple 

 converse problems of Geometry is very evident. The following examples will 

 sufficiently illustrate this remark. 



Ex. 1. To find a curve such, that the area varies as the ?^*■^ power of the 

 abscissa. The general expression for the area of any curve is J' y dx. 



Hence, if ^ = {x) be the equation to the curve, and P 2" the function ac- 

 cording to which the area is to vary, we shall have 



/ 



ydx= P.s" 



