27 
[711 
712] 27 
712. 
LTION OF 
A PARTIAL DIFFERENTIAL EQUATION CONNECTED WITH THE 
SIMPLEST CASE OF ABEL’S THEOREM. 
881), p. 534.] 
[From the Report of the British Association for the Advancement of Science, (1881), ' 
pp. 534, 535.] 
vs. Consider 
tus unity, a 
we have a 
le region in 
60°, and 0°. 
)f x, through 
lage ” of the 
and in any 
e image, we 
by circular 
whole series 
:es or over- 
the size of 
joloured one, 
Consider a given cubic curve cut by a line in the points {x x , y x ), (x 2 , y 2 ), 
(+j, 2/ 3 ); taking the first and second points at pleasure, these determine uniquely the 
third point. Analytically, the equation of the curve determines y x as a function of 
x x , and y 2 as a function of x 2 : writing in the equation 
x s = + (1 — X)#0, y 3 = \y x + (l -\)y 2 , 
we have A, by a simple equation, and thence x s ; viz. x 3 is found as a function of 
x x , x 2 , and of the nine constants of the equation. Hence forming the derived equations 
(in regard to x x , x 2 ) of the first, second, and third orders, we have (1 + 2+3 + 4=) 10 
equations from which to eliminate the 9 constants; x 3 , considered as a function of 
x x and x 2 , thus satisfies a partial differential equation of the third order, independent 
of the particular cubic curve. 
To obtain this equation it is only necessary to observe that we have, by Abel’s 
. , aw + /3 
into , 5,- 
ya> + o ’ 
combination 
theorem, 
dx x dx 2 dx» . 
1 -1 _L i — A 
TT T y T y V, 
^1 ^ 2 3 
where X x is a given function of x x and y x , that is, of x x ; X 2 and X 3 are the like 
functions of x 2 and x s respectively. Hence, considering x 3 as a function of x x and x., r 
we have 
dx 3 X s dx s X 3 
dx 1 X 1 doc2 2 
4—2