504 ON A FORMULA FOR THE INTERSECTIONS OF A LINE AND CONIC, &C. [371 
and thence 
and consequently 
dn7 — 
© 
ydz — zdy = ©P, 
zdx — xdz — ©Q, 
xdy — ydx — ©P, 
ydz — zdy _ zdx — xdz 
xdy — ydx 
ux + vy + wz (nx + vy + wz) P (ux 4-vy + wz) Q (ux + vy + wz) P ’ 
or selecting the value 
zdx — xdz zdx — xdz 
dn7 = 
and writing 
we have 
(ux + vy + wz) Q (ux + vy + wz) (hx + by + fz) ’ 
- = X, - = F, 
z z 
z-dX 
dm = 
(ux + vy + wz) (hx + by + fz) 
dX 
(uX +vY+w)(hX + bY+f)’ 
where X and Y are connected by the equation 
(a,...][X, Y, l) a = 0, 
that is, 1 is a given quadric radical function of X. Hence integrating and restoring 
for m its original value, but writing therein - = X and - = F, we have 
[ d A 1 (a, ...jfo, ft, *&X, F, 1) 
J (uX+vY+w)(hX + bY+f ) V{- (A, ...$>, v, w) 2 } g uX + t>F + w 
where, as just mentioned, F is a given quadric radical function of X determined by 
the equation 
(a, b, c, f g, K§X, Y, 1) 2 = 0, 
and the constants x 1} y x , z x are such that 
(a, y u z x ) 2 = 0, 
ux x + vy x + wz x = 0, 
the ratios of these quantities being therefore determinate ; there would, it is clear, be 
no loss of generality in assuming z x = l. This is Aronhold’s Theorem I. 
2, Stone Buildings, W.C., October 23, 1862.