228 Dr Baker, On the differential equations 



Hence the expression 



ru ru ru 



H (u) = — | Jfi 4 I <f)(u)du— I du du [Jf/i 3 </>(it) + |//< 4 </> 2 (it)] 



is infinite near a like log (u — a), but is not infinite near u = {3. 

 Thus the function 



X(u) = e H M, . 



which is definite save for a factor e Au+B , where A, B are arbitrary 

 constants, is a single-valued integral function vanishing to the first 

 order at u = a and the congruent points (in regard to the periods 

 of (j> (it)). We find, however, at once if we put 



P(u) = -^ 2 \ogt(u) 



that P" (it) - 6P 2 (it) = ^/^ - |/i /*4 + M2-P («*)• 



Hence it follows that, in terms of Weierstrass's particular 

 o--function, 



Z(u) = a(u-a; g 2 , gj) e A/*,«"+^»+B 



where -4, 5 are arbitrary, <7 2 = /^o/Ai ~ i IHIH + tV^ 2 and # 3 ' is to be 

 determined so that the periods may agree with those of $(it), 

 which can be shown to be the case if g 3 is the cubinvariant of the 

 quartic. 



Consider now the differential' equations for p = 2. By equating 

 in pairs the partial derivatives of the left sides we obtain linear 

 equations in the functions g^, jp 221 ,~etc., leading by elimination to 

 the determinantal equation for the sixteen-nodal surface given* 

 on p. 516 of Camb. Phil. Proc. Vol. ix. We consider here the case 

 when \ s = and X 6 = 0, to which the coincidence of two roots of 

 the sextic can always be reduced. Then the expanded equation 

 takes a form which may be written (x = @^, y = ^21 > z = Pu) 



4 [x (xz - y 2 ) + l\ 3 xy - ^\y 2 ] 2 



= (\ x 4 — \^x z y + \,x 2 y 2 - \ 3 xy 3 + \ 4 i/ 4 ) (4>x + X 4 ). 



Following the method of integration sketched in the earlier 

 part of the note just referred to we find from the first three of the 

 homogeneous linear equations for jp 222 , |p 221 , etc., the equations 



2x + y 2 + l\ s y 

 #> 222 = fw, #> 221 = fiy, p m = fi — 2x lx — , 



(Pm = /* 



-lXr + 



2 



y (4z + \ 2 ) (2y + jXs) {zx + y 2 + \\ z y) 

 Ix 2x (2x + ^\ 4 ) 



* Where in the third element of the second row - (4x + X 4 ) should be printed 

 instead of - (ix +£\ 4 ). 



