232 Dr Baker, On the differential equations 



where a, £ are Weierstrass's particular functions, a 3 = — i\ 3 , and 

 the invariants of the elliptic functions are 



1 L V \ , , A* A* 2X 2 3 



al * 3V' J 3A 3 27A 3 2 " 



In this case the functions g) 2S , $ 2l , jp n are rationally ex- 

 pressible by 



U2 ~ SX. Ul ~ ~a? Z( au J> @( au i)> #>'0O> 



which are a doubly periodic set of functions, and the quartic 

 surface reduces to 



(xz -y* + Wy) 2 = A # 3 - XjoFy + \ 2 xy" - \y 3 . 

 (iii) In case A 4 = A 3 = 0, then, save for a factor e Au i +Bv * +c , 



a (u 1} u 2 ) = u 2 + I <f> (w x ) cfa< l5 



where m 



■r 



*yx — \it + a 2 £ 2 



so that in general 



/ \ Ai JA,, — 4A n X(, . . . / — . 

 a (u ly u 2 ) = u 2 + -^—u 1 + sinh (« a V A 2 ) ; 



ZA 2 j^A.2 



but if A 2 =0, \ x 4= 0, 



o- Oj, Mg) = w 2 + -? ^ - — Ml 3 , 

 while if A 2 =0, A x = 0, 



o" («!> ^ 2 ) = u 2 + |n/X ^i 2 , 

 and if A 2 4= 0, A 1 2 = 4A A 2 , 



o- (Mj , ita) = u 2 - ^- U x + — e»' v\ 

 Z/Y 2 A 2 



(iv) Another case, corresponding to that in which the original 

 sex tic has two repeated roots, is when \ = \ 1 = 0, but 



X 2 40, X 3 =f=0, A 4 f0. 



Then a (u 1} u 2 ) may be taken in the original form, and 



* K) = [fe + ^^xT 2 ^ C ° Sh K ^ 2> 

 (v) Of this again a particular case is when A, 3 2 = 4\ 2 \ 4 



