S0LID8 ON SURFACES. 159 



wi -hall find (31) 



; itions to which we shall have occasion to refer hereafter. 



To return now for a moment to the general problem. II 

 we add the second triplet of equations (27) together, after 

 multiplying the firsl equation by Z»„ the second by M, and the 

 third by N. there results 



L,U+M,V+N,W = o. 



Substituting in this equation, in place of the six quantities 

 which it involves, their values (15) and (20). and reducing 



by means of the relations (-1), we shall find 



L .(I {a P'+b Q-hc ft)) 



I .d(a /'-+-// Q-hr Ii)\ = o, 



L .d(a J' +b Q+c"R)) 



where /'. Q. ft are the partial differentials, with respect to 

 p. y. /•. of the function 



T = i(Jp -+- B<i ■+- Cr) — (Fqr+ Grp -+- Hpq) . 



which i- nne half of the living forces of the body arising from 

 in motion of rotation. 



By means of equations (20), the relations ( I) and the sub- 

 stitutions (ft), it will he found that the foregoing equation is 

 susceptible of being presented in the following form : 

 \ ol. in. — i \ 



