238 PROCEEDINGS OF THE AMERICAN ACADEMY 



where 



S(p'\<p'fi((>'v 

 m = — 7- ) 



S(\<t>'fJi(p'v + <p'\fJL<p'l' + <l>'\<i>'fi.u) 



^= s^^;:.; ' 



and 



SfA/U^V -|- \<p'fi.v -f- <^'\fj..v) 

 til ^3 — , 



OKfJLV 



Since the function we are considering is self-conjugate, 



<jro = To'- 

 Let us take first the rectangular form 



and let 



then 



qrA = qr«i = — Cj«i, 



qp^ = qa, = — c^a^, 

 cpv = cpa, = — c^ay 



Substituting these values in our formulas, 



— CiCjCgSOlOoOg 





8(01.0^00.^303 + CjOj.O0.r3a3 -f CjOj.CjOi.Og) 



S(— Oj.Oo C3O3 — Oj.C.^Oo.Og — CjOi-ttj-ag) 



And we find for the general equation of solution 



md=^ — CjCa^gS = — (c^c^ -\- c^c„ -\- c^c^)y — (cj + ^2 + ^3) (^i«iS«i/ 



= — (^1^2 + <^i<^3 + <^2^3)r — ( ^1^2 + CiC8)«lS«ir 



— (C2C3 + CaCj) «2Sa2r — (^3<^i + C3C2)a3Sa37; 

 and for the complete solution 



