24 Dr. Booth on the Rectification and 



«• 



, tan/3' . ^, /*2 , r 1 -1 , 



^ =5^'^'"^y„ ''^L {l-^^sin^y}Vl-sinVsin^J - <'*•' 

 Now as the cones are supplemental, 



a H- /3' = -— , /S+a' = — , sin/3' = cosa, sina'=cos/3; 



, tan/3^ tan/3 , ... . ^ , 



^^""^ i^< = t^' ^ = ^, sin y = sin e. . . (24.) 



Making these substitutions in (23.), we find 



TT 



, tan/3 /*2 , r 1 1 , V 



yss ^COSa / 0.0] , (Qn ^ 



tan« c/o ^L{l-e2sin2^} ./l_sin2esin2^J* ^ ^ 

 Adding this equation to (13.), we obtain the very simple re- 

 lation* 



A + 5' = |-; (26.) 



or taking the whole surface S of the spherical conic, and the 

 whole circumference 2' of the supplemental conic, introducing 

 c the radius of the sphere, we obtain the remarkable theorem 



S + cX' = 2c^n (27.) 



Now cS' is twice the lateral surface of the supplemental cone, 

 measured in one direction only from the centre, and may be 

 put equal to 2 U, hence we deduce that 



XII. The spherical base of any cone, together "doith twice the 

 lateral surface of the supplemental cone, is equal to the surface 

 of the hemisphere. 



XIII. Let S' denote the spherical base of the supplemental 

 cone, and L the lateral surface of the given cone contained 

 within the sphere, then from the preceding equations we have 



S + 2 L' = 2 c^TT, S' + 2 L = 2 c^^; . . (28.) 

 adding these equations, 



(S + 2 L) 4- (S' + 2 L') = 4 c2^; . . . . (29.) 

 subtracting S — S' = 2 (L — L'); . . . . (30.) 



or, 



XIV. If any fwo concefitric cones, supplemental to each other, 

 be cut by a concentric sphere, the sum of their spherical bases, 

 together with twice their lateral surfaces, is equal to the surface 

 of the sphere. 



And the difference of their bases is equal to twice the differ- 

 ence of their lateral surfaces. Hence also this other theorem : 



* The discovery of this remarkable relation between the length and area 

 of a spherical ellipse is due to Professor MacCuUagh, to whom mathema- 

 tical science is so much indebted for many new and beautiful theorems in 

 this department of geometrical research. 



