22 INTRODUCTION TO DYNAMICS. [41. 



Similar formulae result when x is taken as independent vari- 

 able instead of R. For a complete surface of revolution ( = 27r 

 so that j/=o, z=o, as is otherwise evident. 



41. In the case of spherical surfaces, although the preceding 

 formulae can of course be used, it is often more convenient to 

 make use of the geometrical property of the sphere that any 

 spherical area is equal to the area of its projection on a cylinder 

 circumscribed about the sphere. 



Thus the area on the sphere contained between two parallel 

 planes is equal to the area cut out by the same two planes from 

 the circumscribed cylinder whose axis is perpendicular to the 

 planes. The centroid of such a spherical area is therefore on 

 the radius at right angles to the bounding planes midway 

 between these planes. 



42. The Second Proposition of Pappus and Guldinus (compare 

 Art. 30). 



A plane area 5 (Fig. n) rotating about any axis situated in 



in its plane generates a solid 

 of revolution whose volume is 

 V=TT^(y-y?)dx, if the axis 

 of revolution is taken as axis of 

 x and j/ are the two ordi- 



F . j j nates of the curve bounding the 



area. On the other hand, if y 



be the distance of the centroid G of the plane area from the 

 axis, we have 



by Art. 38. Combining these two results, we find 



i.e. the volume of a solid of revolution is obtained by multiplying 

 the generating area into the path described by its centroid. 



The proposition evidently holds even for a partial revolution. 



