28 A PARTIAL DIFFERENTIAL EQUATION. [712 
and consequently 
dx 3 dx 3 X., 
dx x ' dx 3 X, ’ 
where X.,, X 1 are functions of x.,, x 1 respectively: hence taking the logarithm and 
differentiating successively with regard to x x and x 2 , we have 
d d . /dx 3 dx 3 \ _ ^ 
dx l dx 2 ® v^'i dxj 
which is the required partial differential equation of the third order. 
This differential equation has a simple geometrical signification. Consider three 
consecutive positions of the line meeting the cubic curve in the points 1, 2, 3; 
T, 2', 3'; 1", 2", 3" respectively: qua equation of the third order, the equation 
should in effect determine 3" by means of the other points. And, in fact, the three 
positions of the line constitute a cubic curve; the nine points are thus the inter 
sections of two cubic curves, or, say, they are an “ ennead ” of points; any eight of 
the points thus determine uniquely the ninth point.