364 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



of the tangent to the generating curve be represented by <p ; and the inclination to 

 the same ordinate of the tangent to the spiral line by <r. We have then given the in- 

 clination A of two planes to one another, and the inclinations (p and (t of two lines, 

 drawn in them respectively, to the intersection of these planes ; whence by a well- 

 known formula of spherical trigonometry, if / represent the inclination of these lines 

 to one another, 



cos / = cos <p cos ff + sin ip sin o- cos A. 



Moreover, if the two adjacent sides of the parallelogram, being elements of the gene- 

 rating curve, and the spiral, be represented by A * and A S ; then since their inclina- 

 tion is /, the area of the parallelogram is represented by A S . A 5 sin /. Now let us 

 suppose the generating curve to revolve, not altering its dimensions, but sliding along 

 the axis ; then 



ff =z ■—■ .-. cos I = s\n <p cos A, and sin / = v I + sin^ A tan- (p . cos <p ; 



also in this case 



A S = y A cosec A ; 



the area of the elementary parallelogram becomes then 



y x/ cosec2 A -|- tan^ 9 , cos ^ . A * . A 0, or ?/ y/ cosec^ A-f^--^AAA0; 



so that the whole surface of the elementary slice intercepted between two planes 

 passing through the axis which are inclined to one another at an angle A 0, is on 

 this supposition, 



y \/ cosec2 A -h ^3 • dy. 



Suppose the integral in this expression to be represented by N^, then N' will become 

 N (as it ought) in that particular case in which, the curve not sliding along the axis, 



A becomes -^. 



Now we may reason in respect to N^ precisely as before in respect to N, and we 

 shall obtain, by the same steps, the same expression for the surface in terms of N^, 

 as was then obtaine<l in terms of N. 



To find the Centre of Gravity of a Conchoidal Solid. 



Suppose the solid included between P C and Q C (fig. 3.) to be divided into an in- 

 finite number of prismatic elements by planes perpendicular to P C, and perpendi- 

 cular and parallel to A ;2; ; and let m r (fig. 5.) represent one of these elements. 



The VOLUME of this element is represented by 



no X ns 

 or by 



— (mn -\- op) np .n s 



