310 Mr DAVIES on the Equations of Loci 



ED = cos $ = sin 6, and the element of the cylindrical surface is 2 sin 6 d 6 : 

 or integrating between the limits and , we have the semicylinder =: 2, 



m 

 , > 



and the whole cylinder above and below the equator = 8. 



XXII. 



We mentioned, that in the use of the formula A = Ml cos <) d 6, 



certain precautions are necessary to be observed. We now proceed to ex- 

 plain them : to do which we shall commence by an example. Resuming 

 the equation (XX, 2), or 



A = (<p sin (f>) + const. 



And take m n. Hence A = <p sin <p -j- const (1.) 



The area here signified is the portion of the surface over which the ra- 

 dius vector PM passes on the surface of the sphere. If, therefore, we seek 



the area within the limits of </> = and (p = -^, we shall have no space 



passed over by the radius vector but what lies within the curve in question. 

 If we take a greater value of <, we shall have a portion of spherical surface 

 traced by the radius vector which lies without the curve ; in short, the resi- 

 due of the hemisphere after the enclosure of branch OM^L has been re- 

 moved. As we proceed onward to TT ^ </> ^: -5-, we find the half of the 



