ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 109 



Section 9. — The roots of d^,v(.nz) are all included in the expression 

 (2.+^-lV4-(2« + .-l).' ^^^ ^ ^ 3 ^3^^_ 



Hence we find 



\ n / 



(j>{z) being an entire function, which has no finite roots. 

 Putting z+w, z + b)' successively for z, we find 



Substituting this expression it is easy to determine the constant, and we 

 obtain 



»-in-i / aio + ftw'> 



, , n(n— 1)— 



O "fl,v\ ^^ / . 



n -nV-C^^) 



This expression may be transformed, and we obtain the four following ex- 

 pressions : — 



/ecu)+ftco'\ 



0,,X*i2)=«0,,<,(2)n,n^ 



6\ o~ 



e 



'• 1 V n J 



~ 6 1, 1^ 



71+1 



When a. extends to all positive values of a less than — g- , and to all values 

 of /3 less than n, which are positive, except when a^=o, and then /3 is to 

 have all positive values less than ^ , exclusive of zero. Similarly : — 



[ 02 / iZtO + /3U> '\ 



L 



2 /aco-h/iw'X 



0- Z i. 



^'fl-. .z l 



f*,,, o(wz) = 00, o^IIa Up . 



1, o~ — "- — ^— 







This transformation (A. D. M. 3. 138) presents no difficulty if wc remember 

 that it is easily deduced from Section 5, 6 : — 





M. l- 



H,V^ 



