238 Mr. R. F. Gwytlier on the Equations fov 



and 



dm — noyoz oy dz 



= 5 l^(n 1 +n,+a J ) ! 



U+^^- 



B¥ , B 



=^- -(n 1+ n,+n e ;, .... (3) 



where the six fl-functions are all arbitrary harmonic 

 functions, given in a form selected to suit equations (a). 



By substituting in equations (a) from equations (2) and 

 (3), we obtain the condition 



m(m + n) d 2 <fti Bjtf? , B 2 <£ 3 



»{3m-n) V ^ + B.r 2 + V "*" 5-* 2 



~^ + W o^ +c - (4) 



By adding the three equations (3), we obtain also 



?^ + 2@ + ^ + ^) = ^( 2fJl + n 4 ) 



m 



of 

 +.|1 (212, + O,) + |J (2fi 3 + n 6 >. (5) 



We are to regard the functions 1? </> 2 , and </> 3 as known, 

 but subject to a condition. Then, from (4) and (5), 



Qn + nXm-n) ^ d 2 H, ^ ^ 



" *(3m-n) V ^ = 3^" + "3?" + "3?" " 2C - (6) 



The arbitrary functions are to be such as satisfy the 

 surface-traction conditions in any particular case. The 

 whole scheme is approximative and to some extent tentative. 

 H is not anticipated thah the requirements will, in general, 

 be capable of being absolutely satisfied, and any case in 

 which complete disagreement arises would appear to be of 



