PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



617 



but 



/'^^"• = lfe^'^(^^) 



d 



— <P(z)=V.z'' 



or 4>{z) = h'Pz"-' 



hence i/ — ip{x) = n'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 ft'om the place 

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

 operations. 



Let, therefore, y = (p{x + a) be the equation to the curve, the limits of integra- 

 tion being x=0 x—s, 



then J^ (l){x + a)dx = V .\ogz by the question 



d- 



or 



dz- 



(f)(z+a) = P. logs 



(p{z + a) = -^F.logz 

 P 



or 



^ = T 



xy=P 



which is the equation to the hyperbola. 



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

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

 Let t/^=(p(z) be its equation ; 



volume ='7rfJ'(p{x)dx 



./■ {x) being the given function of x. 



'TT dz 



"7iJI^-> 



is the equation required. 

 Cor. If /(2) = P^» 



.^=-7— VwP.a"-! 

 = V .z 2 



