018 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Ex. 4. A curve is described having a line of given lengtti as its axis. From 

 the further extremity of the line is described a reversed parabola, having a com- 

 mon axis with that of the curve. A third curve is then described, whose ordi- 

 nate is a mean proportional between the ordinates of the former cm*ves, and such 

 that the volume of the solid described by it between the limits of the line in ques- 

 tion is a certain given function of the length of the line : Required the equations 

 to the two curves ? 



Let t/=z(f)(x) be the equation to the first mentioned curve ; 



z the length of the line, which is made the axis of a; ; 



/{z) the function of z according to which the volume of the solid swept out 

 by the last curve varies. 



Then f=\/m{z—x). (px is the equation to this curve ; m being the latits rectum 

 of the parabola. 



Therefore tt V^y* dz Vz—xcf) (x) is the value of the volume of the sohd swept 

 out ; so that nr Vmf" d x Vz-xcp («) = /(«) by the question ; 



"^ "^ ^ 2 sm?«7r dz ^ 



A being some constant. 



d^- 

 And consequently 9 (a;) = A -—Tfix) is the equation to the first curve. 



dz^ 



The second is immediatelv deducible from it. 

 Cor. 1. Let /(2)=-" 



-—^/{x)=Gx''-i 

 dx^ 



and (p {x) oc a;"-t . 



Cor. 2. If n = 2 (p (x) o: x^- . 



In this case both the curves are parabolas, and the volume of the sohd varies 

 as the area of a circle, whose diameter is the given line. 



I shall now conclude the series of examples. It was originally my intention 

 to have exemplified the theorem of expansion given in my preceding memoir ; 

 but, on consideration, I deem it advisable to confine the present series to the 

 illustration of the theorem which forms the commencement of the paper. I 

 hope at some future period, should no one render it unnecessary, to return to 

 this subject ; and look in the mean time for the fruit which shall be produced by 

 a more extended culture of the science. 



Edinburgh, January/ 20. 1840. 



