ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 335 



Report on Recent Progress in Elliptic and Hyperelliptic Functions. 

 By W. H. L. Russell, F.R.S. 



We no-w enter on the consideration of the Hyperelliptic Functions. I propose 

 to divide the subject into four parts, thus :— 



Part I. On the System of Hyperelliptic Differential Equations adopted by 

 Dr. Weierstrass. 



Part II. On the System of Hyperelliptic Differential Equations adopted by 

 Jacobi, Gopel, and Rosenhain. 



Part III. On the Transformation of Hj^erelliptic Functions. 



I hope to add 



Part IV, On certain Theorems not involving the Periods of the Functions, 

 with a Supplement to the Report. 



Paet I. On the System of Hyperelliptic Differential Equations adopted by 



Dr. Weierstrass. 



We now proceed to explain the discoveries of Dr. Weierstrass. It will be 

 seen that the form of his hyperelliptic differential equations is different from 

 that assumed by Jacobi, Gdpel, and Rosenhain. The object of Weierstrass is to 

 solve these equations ; and the advantage of his method will be seen when 

 we consider that he solves the hyperelliptic equations generally, and not for 

 ■a particular case, which is aU that Gopel and Rosenhain had previously 

 effected. Weierstrass assumes as follows (CreUe, 47) : — • 



doa 



^ ^ r""' f (^) f?^ p P(a;) dx C'^n T(x) 



-«/2'V/R0r)' 



) tZ.-c 



&c.=&c. 



_p P(.r) dx r^' P(a;) dx r'n P(^-) dx 



''" Ja,^'-«2.-i'2^R0^ J^^-^-«2„-r2^/R(^) •••• J,^_^^-«2„-i'2VE(^) 



where ^(x)=(x-aj(x-aj(a;-a^) (*'-«2«)' 



■e(x) = (x-aj(x —a,).. . .C^'-«2,^-l) > 



and let Q(x)=(x—ao)(^'-«2) (^— «2»)' 



so that lEi(x)=V(x).Q(x). 



If L(x) = (x—x^){x—xj (^— a?„), 



we define 



7/ N V(-l)«L(a) 



al(u^u,....uj^= : \y[ , 



^(_1)-R'(«J 

 where d is the greatest number contained in la ; 



