255 



It is then shown, by the aid of the auxiliary ellip=!oi(l, whose 

 equation is 

 (a-lK + (b-m)//2 + (c-N)2r^ + 2(a, + /3, + y,)yz 

 + 2(a2 4- i32 + y.:)xz + 2(a3 + jSs + y:,)xy =1, (6) 

 that the function Vi and the surface (5), may be referred to a sys- 

 tem of rectangular axes, for which the following relations exist : 

 S. +n, + 2, = 0; S, + D, + 1, = 0: S3 +23 -1- 2, = ; (7) 

 the Hebrew letters denoting what the Greek become after 

 transformation of coordinates. The possibility of these equa- 

 tions in every case amounts to a proof of the existence of three 

 axes at each point of a body, which are intimately connected 

 with the molecular constitution of the body round the point. 

 The equation of equilibrium of a solid body is then shown 

 to be 



SSS(xS£+yg,,+zS?)c?m= A -\ll{v,ll+Qi,^n-\-K,lZ,)dxdydz, (8) 

 where 



d'y\ ,0 d'r, d'r, f d'n d'n „ d'n \ 



^ dX_^^d^t, d't^J d'Z _^ d'Z ^^d^Z 



"- •! ■ "- <i '^'*J ^/^^ d'n , ,, (?''ij ^ d'-n \ 



d^n , fZ^rj C?-w ^/^^ d'ri . ^, (?^r 



dy- dz^ ' rfa;' 



rf^'^ , d% . (^^^ „^^ d% d% d% \ 



,r,d%, d^Z. d% ( d% d% d%\ 



_ d% d^Z d't. Jr. d'Z d^l d'l \ 



d% , cZ^^ , (^'^ , _ / d-l , ^t , d% \ 



J^g d^l,P>d'l,J d'l ,^d% rf^^N 



VOL. HI. 



