The Rev. J. II. Jellett on the Equilibrium and Motion of an Elastic Solid. 1 9.3 



Since then these equations are supposed to liold for all possible displace- 

 ments, we must have the six equations 



Six equations being furnished by each of the remaining equations (S), we 

 shall have in all eighteen equations which are obviously identical with (R). 



9. If the sum of the internal moments admit of being represented by tlie 

 variation of a single function, the tliree directions of molecular displacement 

 corresponding to a given wave plane will be at right angles to each other. 

 This has been shown by Mr. Haughton. AVe shall now proceed to prove the 

 converse of this theorem, namely, 



Tf the three directions of molecular displacement corresponding to the same 

 wave plane be at right angles., and if this be true for every wave plane, the sum of 

 the internal moments of the body may be represented hy the variation of a single 

 function. 



We have seen that the directions of molecular displacement corresponding 

 to a given wave plane are determined by the equations 



— ev^ cos ^ = n, cos /+<!>, cos m + "^i cos n, 



— ev- cos ffi = lis cos I + (^2 cos m + ■*'o cos n, 



— ev- cos n = ris cos / + <I>3 cos m + ^3 cos n. 

 Eliminating ev^ between the first two of these equations, we have 



n, cos I cos ?« + Oi cos- 7n + ^, cos m cos n 

 = Ui cos- / + <I>2 cos / cos m + ^2 cos I cos n. 



VOL. XXII. 2 D 



