3174 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
Also the invarlantal coordinates are connected with the ordinary coordinates by 
relations having the form distinctive of a homographic transformation. And there¬ 
fore, regarding the invariantal coordinates as ordinary coordinates, the curve 
represented by the intrinsic invariantal equation has all the perspective singularities 
of the original curve. No attempt is made to show the application of this method to 
the difficult case of the cjuartic, but several well known theorems regarding the 
cubic are shown to be arrived at without difficulty. 
1. In general, the complete curve of the algebraic degree is completely deter¬ 
mined by (cZ + 3)/2 {=r) points, and if the coordinates of these points are known, 
the equation to the curve can be written down in the form of a determinant, thus 
f{x, y, . . .) = 
... 1 
x'^~hj, ... 1 
/yi (I /yt (1“" ],■) I f 
yi? • • • 
= 0 
If a general homographic transformation of the coordinates be now made, the form 
of the function will be unaltered, except by the introduction of a factor. 
Denote the trajisformation by 
?/ 
AjX + BiY + C'l AoX + iqY -h C3 AX + BY + C 
with similar relations for ^ and rj, except that we may retain the letters t], in the 
transformed equation without causing inconvenience. 
Thus 
f(x, y, . . .) = D + (Af + B, + C)-" (AX, + BY, + C)-'' 
X/(X, Y, ...). 
We may also write 
f{x, y, . . .) = 
- ocY, - xY 1 {rj - y), 
— xY, {xj - xY~^ {yi — y), 
^ — sc, y — y, 
Vi — 
Imagine the r points now moved up into coincidence at x, y, subject to some 
