616 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION 



di 

 — —J- ^ (z) = a const. 



zi <i>{z) 



and <P (s) 



A suppose ; 



A 



or 



rf* _ A 



the differential equation to tlie cycloid. 



Lest any difficulty should he 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 oiu- steps : 



^2^^ > l-^ -sinlTT rfs-i^^^ 



/ = 



since m = -^ p — -^: 



hence, if a be the constant time, we get 



,,, \/2o -1. di 



Jit*/-! sinOTT dz' 



_ '^'2.g _ —1 sinOTT j a 



^7r\/ — 1 sinO'TT Jtt 



^/ 



z 



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



ordinary process of expansion. 



The facHity 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, it y=(p(z) be the equation to the curve, and Pz" the function ac- 

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



f: 



rf.r= P 



