1898.] differential equations defining periodic functions. 



(a) Putting 



f (x) = X + \x + \.m" + \ 3 x 3 + \ 4 x* + \ 5 x r ' + X 6 ic 6 , 



F(x, z) = i x'z 1 [2X 2i + \., in (x + *)], with X 7 = 0, 



and ^ = x, p a = y, p u = 2-, 



the equations are 



- 2 (^2222 - 6^1) = ^e - 1 X3X5 - '2K»' - 2\ 5 2/ + ( ^«^ 



- 2 (^., 2] - 6^^) = %\\ 6 - 2X 4 ?/ + V 

 -2(g> 2211 -2g> 22 ^ 11 \= \ X 6 - \ 3 y 



517 



~ 4 $i 



- 2 (jp 2U1 - Gp^u) = |XoX s + X^ - 2X# 



- 2 (p im - 6p„) = X X 4 - i W + G\ x - 2\y - 2\. 2 z, 



which take a still simpler form when we put X 6 = 0, X B = 4, as we 

 may do without loss of generality. By differentiating the first 

 equation in regard to u x , and the second in regard to u 2 , and sub- 

 tracting, and acting similarly for the other pairs, we obtain four 

 equations, of which, for instance, one is 



- Syftn + 2acp m + |X 5 g> 2 „ - \#m = 0, 

 and thence by elimination we obtain 



— 2y 2x |X B — X 6 = 0, 



2z 2y + %\ 3 -(4aj + iX 4 ) i^s 

 |X, -(4^ + X,) 2y + ±\ 3 2x 



-X JXi 2.2 -2y 



which geometrically is a transformation of the equation of the 

 sixteen-nodal quartic surface first found by Gopel and considered 

 by Kummer. The nature of the transformation is best described 

 geometrically — suppose X fi = 0, X 5 = 4, and use the abbreviations 



P b = y + bx- b\ P bc = z + (b + c) y + bcx- e b)C , e b<c = -^ „ J , 



then, if d, a x , c 2 , a 2 , c are the five roots of the equation f(x) = 0, 

 and P denotes the plane at infinity, the sixteen nodal points are : 



I. The infinite end of the axis of z, through which pass the 

 six singular tangent planes 



P P P P P P- 



