176 Dr. J. Bottom LEV on 



The whole surface will be 2Trx^-\-2ir.xy,.v denoting the radius 

 of the cylinder and j its length, also the volume will be 

 Tra-;'. If the cylinder be isotropic, and dv the decrement of 

 the radius, dy the decrement of each extremity, we shall 

 have the relation dx=-dy\ whence j/ = ,r+j/o—,t,„ and the 

 expression to be integrated become 

 dx 



■KJ? + izx\yo -Xo) + r 



dx 

 ■KK? + Trar(jo - x„) - r 

 xd 



-Idt, 



- kit, 



- kit. 



TT^ + Trx-{yo - Xg) 



Hence in each case the velocity of the action is expressible 

 as an algebraic function of the variable x ; in each case the 

 determination of the complete integral will offer no diffi- 

 culties when the arithmetical values of the constants enter- 

 ing into the equation are given. If the length of the 

 cylinder be equal to the radius, the differential equations 

 differ from the corresponding equations for the sphere in 

 haviner tt as the coefficient of x'' instead of — , and the 

 integrals may be obtained by making this substitution in 

 the corresponding integrals relating to the sphere. 



As another example, suppose the solid to be one of the 

 regular solids ; then x denoting the length of the edge of 

 one of the plane faces bounding the solid, for the volume of 

 the solid we may write w,^-^ and for the surface nx' ; the para- 

 meters jn and u having different values for each of the five 

 regulai polyhedra. Differentiating the expression for the 

 volume with regard to x, and substituting in the general 

 equations of solution we obtain 



5^^^-=-lndt 

 mx^ + r 



^^^f' =-lndt 



mor - r 



^^=-lndt, 



