of the hyper elliptic functions. 



231 



where 



log H (^) = \ v% I </> W d«i 



-•«, 



and log i^O differs from this in the sign attached to v\ 4 . Hence 

 if a, /3 be incongruent poles of <£ (u), such that 



4>' (o)/f(a) = VV, 



f( i 8)/^(/3) = -Vx;, 



it follows from what we have proved above, that 



o- (^ , Us ) = [e? u > «** o- («! - /3) + e - *^ \/^ 4 g. ^ _ a ^-j e 24 



where a (u) is Weierstrass's particular function depending on the 

 two invariants 



g 2 = X X 4 — \ X x A, 3 + Y2 V» 



#3 = i ^-0^-2^4 + ^g^i^X 3 — ygX/Xj — T 1 g-A, 3 2 X — 2J6 X 2 3 . 



It is easy by differentiation of this expression to find the 

 functions (jp 22 , jp 21 , g? u and to show that they are rational functions 

 of the three 



it- a (u x — 8) , N , , N 



which are a set of triply periodic functions. 

 It may be remarked that the equation 

 <r(u 1} w z )=0, 

 regarded as an equation for determining u 2 in terms of u 1} gives 



tiri 



Vx? 



u 2 + I ^ (?(i) cfoti 



where n is an arbitrary integer. 



(ii) When X 4 = and the other coefficients are general 



o- (u lf u 2 ) 



< 2 + I (Wi) 



C^U, 



fr 3 f Ul du 1 j Ul <Hu 1 )dui 



is easily found to be, save for a factor e Au i +Bu * +c , 



(u 1} u 2 ) 



x 2 1 yt ' 



(a/Uq) e 



