ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
75 
coordinates of the consecutive ineunt will be {x-\-dx, y-\-dy, z-\-dz)^ and the line joining 
these two points will be the tangent to the curve at the point {x, y, z). Take (X, Y, Z) 
as current point-coordinates, the equation of the tangent is 
X , Y , Z 
X , y , z 
-0, 
x-^dx, y-{-dy, z-\-dz 
or what is the same thing, 
X( ydz — zdy) -{-Y{zdx— xdz) + "Li^xdy —ydx) ~ 0. 
But since U is a homogeneous function of {x^ y, z), we have 
x'b^V =mU = 0 ; 
and since {x-\-dx, y-\-dy, z-\-dz) is a point of the curve, we have 
dx'b^JJ + dy'b^'U -f dz'b ^ = 0 ; 
and from these two equations 
ydz — zdy : zdx — xdz : xdy — ydx=^JJ : : B^U, 
and the equation of the tangent consequently is 
XB,U+YB,U+ZB,U=0. 
190. Take (|, Q as the line-coordinates of the tangent, then the equation of the 
tangent is 
|X+;,Y+^Z=0; 
and comparing the two forms, we have 
and if from these equations and the equation U = 0 (the point-equation of the curve) 
we eliminate (^, z), we obtain an equation between (|, pj, t)^ which is the line-equa- 
tion of the curve. We may, if we please, present the system under the form 
B^U + ^l = 0, 
Bj,U -f = 0, 
U= 0, 
or what is more simple, under the form 
B^U+^l=0, 
B3,U+Xp?=0, 
B^U -1-A^=0, 
lx-\-r,y-\-lzz={i, 
and from either system eliminate x, y^ z and 
