1898.] differential equations defining periodic functions. 519 



The expressions for u,,, w, as linear integrals can now be 

 obtained as in § 1. The result can be brought into a familiar form 



Ldx -f Mdy (Mdx + Ndy 



rLdx + Mdy f 



1 d£ J 



where ^ = is what the equation of the sixteen-nodal quartic 

 becomes when we substitute for z in terms of x, y and £, where 

 (f= $->z-i, and L, M, iV are rational integral polynomials in x, y, £. 

 From the point of view of the theory of functions it is the surface 

 ty = 0, rather than the sixteen-nodal quartic, which is to be 

 regarded as fundamental. 



(/?) Coming next to a certain symbolical form of which the 

 differential equations are capable, we notice that we have 



and similarly 



g>2222 - 6^ =.- - — ^— = - ^ A 2 o-o- , etc., 



where 



Alacr' = ( - - r— 7 J o- (Mi, « 2 ) <r (it/, m 2 '), etc., 



\OU.j OU-2 / 



the variables u{, m 2 ' being replaced after differentiation by w x , m 2 . 

 Now let k.,, k 1} X, fi, X', [jl be arbitrary constants, let us take 

 Klein's invariantive form for the polynomial F {x, z), and 



x,a f x (Xx + /J,)dx ,.,„_ [ x (\'x + fl')dx j fi 6 



then the differential equations are summed up by 





= (\fi' - X'fi) 4 . (<f>, fY (fy k fl . aa 



—{Xfjf — X'/x)' 2 . <j> k 



d . d\ ( . d , d 



^^r^J'v'dv.r^dv,' 



where a is regarded as a function of v 1} v 2 , and, after differentiation, 

 v./, Vi are to be replaced by v 2 , v t . 



This form shews, by equating terms of the same dimension on 

 the two sides, that the equations can be used to obtain the expan- 

 sion of the c-functions in integral powers of the arguments, and 

 gives an immediate proof of the known theorem as to the 

 invariant and integral character (in the coefficients of <j>l) of the 



