XXX IV 



Trans. Acad. Sci. of St. Louis. 



(5) ll^ = C7— ^ 



k L 



A,J,(^,r)+BjL\(^,r) 



+ 



k 



l^v, 



A^.(v,r) + Blli,(vjp-) 



2{2/ji + X) 



2 



^• L 



-llM(v)+^]^^^o(^;fcO 



'Jk^^^-h'^ 



^Vk^ j^^-V,z^ 



(e^*^+e-^^^) 



give a general solution of the equations (2) where the coeffi- 

 cients A, B, A^, B^ are determined from surface conditions. 

 For the purposes of this paper, however, a simpler method 

 may be pursued. Namely, formulas (4) and (5) show that 

 Jthe general solution may be presented in the form 



(6) 



«^i =«*io + ^^n ^^ + "i2 «* + 



where m^o, w^, . . . , ?<^o> ^^i» • • • ^^e functions of r only. 

 Substituting these expressions of u^ and ly in (2) we find that 

 this system of differential equations may be replaced by the 

 following jone 





fjt' d / dii\ 





'CZW,, M, 



/cm,, M,,\ 

 (8) +2(M + M(sr+r") 



= 0, 



= 0, 



