ON ELLIPTIC AND HYPEEELLIPTIC PUNCTIONS. 353 



are connected linearly with ?(j«., &c., and ^civ^v.^ . . . . ) can be expanded in a 

 series of exponentials. Moreover, -we now see that A\{u^u.,. . . . )^ depends 

 on J^(i/j— JH^JT+Sji. . . .)) ■n'hen 5^{i\—m^n-\-c^i. . . .) can be expanded in a 

 sci-ics of exponentials. Hence cil{u^u^. . . .)» ^^^ ^^ expressed as the ratio of 

 two scries of exponentials, a theorem equivalent to the well-known sinawi 

 2Kj;_ 1 e^x 



T ~~ VI- ^'^'' 



Section 14. — We shall conclude this part of our subject by giving the ex- 

 pansion of hyperelliptic functions in terms of divided arguments as given in 

 Dr. Weierstrass's second paper. 



Let ll(a7)=(a?— flJ(.^•-a,) (•'''— '''■.p+i)* 



F(x) = (x - f? J(.v - ffj (.r - Wp), Tx{x) = 'B(x)Q,x, 



Q{x) = (x--.%\)(^v-x^). . ..{x-.v^), 



7 _1 ^(^ J^mI l^ ^^4- 4-IifpI J^ 



,;„_2 I(^il ^^ , 1 I^ _i^. , ^'^P ^^'•'•p 



f - p 



" 2 a^^ - «^ V Ra-^ "^ 2 .^•^-a^ ' V E.r^ '^ " " '^ x ~n ' Vj^ 



p 



ix^ 



Any one of these equations may be written 2 - . m . ji_ __ du 



where 2 applies to ^, and extends fi-om 1 to p. 



Now let .<, .<. . . ..r;, <', <' . . . .xj' .r/'">, .i'/»0 .r/'") be a 



set of mp variables corresponding to the arguments tf^', uj . . . .u ', v " , «./' 

 .. . .»", «/'"\ ef/"'\ . . •^'p"'^ ('« being an even number), so that 



1 P.*'' f?.*-!, , 1 P.r" f?,r" 



^^ -V-^i VR,r^ " ^-.r^-^, VR/ ^ ' 



^2 .r(;")-«^ VRri^'"^ ^ ' 



1 p-'v ^< , , 1 P.*-; ^K 



"i 





Ve.i/' 



;x M - -^ • > 



1872. 



1 Pa-^'") rfa/'") 

 itc.=iS:c., 



vi . ^^ .- 1> f],,On) 



-^u "■■ VR.r'"') 



i; B 



