352 REPORT— 1872. 



Moreover 



a 2J0<-4.K) 



A1K-K,....) = e " " ''Al(«,....)„ 



A1K....)„. 

 Combining these results we have 



where U= -S.c^M^+iSe'^M,- | Vc^^+iVr'-'c- J Vc<^o+ ^^\'>n^'\- |2,e> 

 - Pe>',+ 1 Va2;,+ Se.t*,- i2e>.- |2,e,<^,+ J 26,..',+ 1 2,e>,+ ^Se'.o,',. 

 Now 2,e>e- 2,6,0,; = 



Consequently, substituting this in the preceding formula and reducing, we 

 shall have 



U = — 9 ^x.^xi'"?,. ~ I '"a"" + 2^A^) ' ^^^ therefore 



g.e 





which is the formula 60, p. 305. 



Weierstrass then shows that we are able to expand J^Ct^i— WjTr + S^i. . . .) 

 in a series of exponentials. 



Section 13. — The theorems just given contain in fact the solution of what Clebsch 

 and Gordan have called the ' Umkehr Problem,' as applied to the hyperelliptic 



functions; for we have already given al(u, u . .) = — ' 2 ■ • /g . ^^^^ 



" Al(tijtt„ . . ) 



we have also shown that Al(u^u.^ . . . . ) depends on JpC^'i^a ....), where v , v. 



