PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. QIJ 



but X-^'^"'"^'^^"^ 



0(^)=P, 



dz' 



or (p (z) — nP z"-' 



hence ^ = («) = « P . x"-' is the equation to the curve. 



Ex. 2. To find a curve such that the area shall vary as the logarithm of the 

 abscissa. 



In this case, we must suppose the origin to be at a distance from the place 

 at which the area commences, in order to prevent the appearance of go in the 

 operations. 



Let, therefore, y = (j)(x + a) be the equation to the curve, the limits of integi-a- 

 tion being x=() x—z, 



then J^^~ (p(x + a) dx — P . log z by the question 



d~^ 



or -—^(p{z+a)=^P.\ogZ 



(pi^ + cc) = -^P.logZ 



JP 



z 



p 



or j/ = — 



X 



xy = P 



which is the equation to the hyperbola- 

 Ex. 3. To find a curve such that the volume of the solid generated by its re- 

 volution round the axis of x shall be a certain function of oo. 

 Let f=(p(x) be its equation ; 



volume ^TT fj' <p (x) d X 



./' (x) being the given function of x. 



'TT dz 



is the equation required. 

 Cor. If /(^ = Pz" 



1 , 



j/=-7— V»P.««-i 

 / 7lP !t± 



= V .Z 2 



TT 



