98 
[818 
818. 
NOTE IN CONNEXION WITH THE HYPERELLIPTIC INTEGRALS 
OF THE FIRST ORDER. 
[From Grelles Journal der Mathem., t. xcvm. (1885), pp. 95, 96.] 
In the early paper by Mr Weierstrass “Zur Theorie der Abelschen Functionen,” 
Crelle’s Journal, t. xlvii. (1854), pp. 289—306, we have pp. 302, 303, certain equations 
(43), and (stated to be deduced from them) an equation (49). Taking for greater 
simplicity n= 2, the equations (43) written at full length are 
K\\ J12 J\ 1 -f* K%1 JJ2 -Ä-22 J%1 9, 
(43) ■ K U J' 12 — K 12 Ju + K 21 J'22 ~ K 22 «/21 = 0, 
K n J 11 A \\Ju + A 2 i J 21 K 21 J>i — V’U, 
K' n J\2 - K’ V1 J' 11 + K. M J'^- K'vJ'n = 0, 
A 12 /'ll - K' U J 12 + K,2 J'21 - ZW22 = 0, 
Z 12 /' 12 - K' 12 J 12 + K.,2 J r 22 - ; 
viz. in the theory of the hyperelliptic functions depending on the radical 
V# — a 0 . x — «1. x — a 2 • x — a- t . x — a 4 , 
these are relations between the eight integrals K of the first kind, and the eight 
integrals J of the second kind. Each equation contains both K’s and J’s, and there 
is not in the paper any express mention of a relation between the K’s only, which 
occurs in Rosenhain’s Memoir, and is a leading equation in the theory. But taking 
as before n = 2, and for the G’s which occur in (49) substituting their values as 
obtained from the preceding equations (46) and (47), the equation becomes 
(49) K n K'v - K 21 K’ n + K 12 K'v - K^K\, = 0, 
which is the equation in question : it is the equation o) 0 v :i — o) :i v 0 + — eooVi = 0 of 
Hermite’s Memoir “ Sur la theorie de la transformation des fonctions Abdliennes,” 
Comptes Rendus, t. XL. (1855). 
It is interesting to see how the equation (49) is derived from the equations 
(43). I write for greater convenience 
K u , K!2, K21, A22, K11, K 12 , K 21, K 22, J\\i /12» J21 ) /22» J11) J12) J 21) J 22 
— A , B , G , D , A' , B' , C' , D' , a , /3 , y , 8 , a' , /3' , y , S'.