of the hyperelliptic functions. 227 



or, if fp + iV^a = t, 



where g 2 = A-o\i — i\~^i + 1*2 V s 



is the quadrin variant of the quartic form 



\ + Xi# + X 2 # 2 + \ 3 # 3 + X 4 iC 4 , 



but <7 3 ' differs by an arbitrary constant of integration from the 

 cubin variant 



( Ji = i^O^-2^-4 + 4£ A-iX 2 \ 3 — Jq\ 2 \ — yg-X 3 2 A — TTS^' 



we infer that the general integral of the differential equation is 



- 1 \\ 2 + {p(u+ct, g 2 , g 3 '), 



where a, # 3 ' are arbitrary constants, <@ denotes Weierstrass's par- 

 ticular function, and g 2 is, as above, denned by the coefficients in 

 the differential equation. 



There is, however, another form of integral naturally suggested 

 by the following work for p = 2, which though not new seems 

 worth recapitulation. 



Let $(w) be a definite single-valued function satisfying the 

 equation 



fdxV 



I -T- I = f>h + frX + f^OT + fA 3 %* + /i 4 tf 4 , 



wherein in the first instance we suppose the coefficients on the 

 right to have general values and ^ not to be zero. At a point at 



which <f> (u) is infinite the limit of -— - may have one of two values ; 

 denoting a definite square root of /x 4 by J ix A , let 



f («)_ ,- 0'O8)_ ,- 



Then near a we find 



(u) = ~- + positive powers of u, 



where v = — sj~/l 4 (u — a); 



near /3 we have the same expression in terms of 

 w = + 7^4 {u - /3). 



