71 



Therefore 



because OB and OR are equal. 



Imagine a sphere described from the centre O with a radius equal 

 to OB, and passing through R' and S', and cutting A'O in P. The 

 sides BP and PR' of the spherical triangle BPR' subtend the angles 

 BOA' and A'OR' at the centre ; and its angle PBR' is equal to the 

 angle ROT ; whence the sines of the sides being porportional to the 

 sines of the opposite spherical angles, it follows that 



sin. BOA' 



sin. A'OR' 



sin. ROT 



is equal to the sine of the spherical angle BR'P, which is the angle 

 made by the section A'OC with the plane of the circular section 

 BOR. Similarly, by means of the spherical triangle BPS', it may 

 be shewn that 



sin. BOA' 



sin. A' OS' 



sin. SOT 



is equal to the sine of the angle made by the plane A'OC with the 

 plane of the circular section BOS. Therefore, since the angles 

 A'OR' and A'OS' are equal, it follows that 



— TTTT >s equal to 



OC" OA' ^ DC' OA' 



multiplied by the product of the sines of the angles which the plane 

 A'OC makes with the planes of the two circular sections. 



I shall now demonstrate a geometrical construction for finding the 

 magnitude and direction of the elastic force arising from a displace- 

 ment in any direction — a construction, which, with the help of the 

 preceding lemmas, will lead us immediately to all the conclusions 

 established by Fresnel. 



