Medium, according to the Principles of FresneL 7 



Imagine a sphere described from the centre with a radius 

 equal to OB, and passing through R' and $', and cutting AO 

 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 proportional to the sines of the opposite spherical 

 angles, it follows that 



sm A OR 



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 shown that 



. 

 -, sin SOT 



sm A OS 



is equal to the sine of the angle made by the plane A'&6' with 

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

 angles AOR f and AOS f are equal, it follows that 



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 find- 

 ing the magnitude and direction of the elastic force arising from 

 a displacement in any direction a construction which, with the 

 help of the preceding lemmas, will lead us immediately to all 

 the conclusions established by FresneL 



Let be the position of a point in the medium when 

 quiescent, and let three rectangular axes passing through it, and 

 fixed in space, be taken for the axes of co-ordinates. For a small 

 displacement in the direction of x, let the elastic forces, excited 

 in the directions of x, y, s, be a, b, c ; for an equal displacement 

 in the direction of y let the forces be #', b', ,c ; and for the same 



