618 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Ex. 4. A cm-ve is described having a line of given lengtli as its axis. From 

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

 mon axis with that of the cui-ve. 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 tlie two curves ? 



Let yT^(p{x) be the equation to the first mentioned curve ; 



~ the length of the line, which is made the axis of x ; 



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

 by the last ciuwe varies. 



Then /=\/»i{:—x). (px is the equation to this curve ; m being the lattis rectum 

 of the pai'abola. 



Therefore tt Vm/^ dzV:-x^ (x) is the value of the volume of the solid swept 

 out ; so that tt Vm/yxVz-x(p («)=/(«) by the question ; 



as - 



A being some constant. 



rf = 



And consequently ^ («) = A -—-jf{x) is the equation to the first curve. 



The second is immediately deducible from it. 

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



di 



—-^/(x)=Cx'-^. 

 dx- 



and (p (x) oc x''-i . 



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



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

 as the ai'ea of a cu-cle, whose diameter is the given Mne. 



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 foi-ms 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. 



