and the Rectification of Curves. 211 



and therefore 



p _ r 2 sin0 



V{r 2 sin 2 + sin 2 0(D e r) y +(lV) 2 }' 

 4. As an example of the application of the formula for the, 

 quadrature of surfaces, let us suppose that it is required to inves- 

 tigate the quadrature, between given limits, of the surface 



r=j»e~* cos 0. 

 Then 



D e r= —?ne~*' sin 0, D(pr= —me~^ cos ; 

 therefore 



da= \/(m*e-*?cos*0sm*O + m' 2 e- 2 tsm 4 0-i-7n*e- 2< i i cos*0)rddd<p, 

 or 



da- = n?e-^ cos d0 d<p ; 

 whence 



S = m 2 / e - 2 *(sin6' 2 - siii0,)£ty. 

 Let us suppose the limits to be given by the intersections, with 

 the given surface, of the cones 



2 = a<p, 9 x -=b<f>, 

 and 



2 = m 2 \ e -2 ^(sin «<£ — sin bcj>) d<f>, 



an integral which is susceptible of easy reduction, since we know 

 that 



i 



_. . , ,, - m sm ad> + a cos ad> 



-ms sin «<f> </<£ = — e _m * V^— 5- -« 



r r m 2 + e 2 



5. As a second example, let it be proposed to investigate the 

 quadrature, within given limits, of the surface 



r=mcos(j) sin 0. 

 Here 



D e r = m cos </> cos 0, D^r = — m sin (/> sin 0, 

 and 



da=-irv l cos <£ sin 2 d#c?</> ; 

 whence 



2 = m 2 1 (sin <£ 2 — sin 0,)sin 2 #J#; 



and, if the limits be given as before, there is no difficulty in de- 

 termining the quadrature completely. 



6. In the masterly treatise upon the Calculus of Variations by 

 the Rev. Professor Jellett (Dublin, 1850, p. 262), it is shown 

 that the surface which, within given limits, renders the double 

 integral 



P2 



