CONTACT AT ANT POINT OP A PLANE CUEVE. 395 
ing equation contains the equation of the tangent as a factor, and rejecting this factor, 
the reduction in degree is two units. 
49. Applying the method to the two equations, T=0, 2W— nV=0, and substituting 
therein for the original current coordinates X, Y, Z the values A.'r+f/.X, 
Aj+jO/Z, the equations become 
A^U + 3a>V H- 3a^^ 0, 
2(A^U+2Aja,V+^^W)-(AP+|f^nXAU-f/4V)=0; 
or writing U=0, and omitting from each equation the factor |M», the equations become 
AW+A/a.3W+^^T =0, 
A(4 - P)VH-^(2 W- rrv) = 0 ; 
and putting in the first equation A=2W~nV, ^=_(4~P)V, the result of the elimi- 
nation contains the factor V, rejecting which it becomes 
3(2W-nVX-3(4-P)(2W-nV)W+(4-PXVT=0, 
which is of the fourth degree in (X, Y, Z), as it should be, and represents therefore 
the lines drawn from the point of contact to the other four points of intersection of the 
conic and cubic. 
50. The equation may be written 
_ 3(2 W - nv)((2 - P) w-f 11 v) -f (4 ~ PXVT=: 0, 
and we obtain at once the condition that this may contain the factor V, viz. this con- 
dition is 
P=:2; 
and if this be satisfied the conic will have a three-pointic conic, and there will be three 
other points of intersection. And writing P=2, and throwing out the factor V, we find 
3n^V-6nW-|-4T=0 
for the equation of the lines from the point of contact to the three points of inter- 
section. And we have now to determine II so that the function on the left hand mav 
divide by V^. ^ 
51. I simplify my original method by the use of a theorem of Mr. Salmon’s, viz. 
writing 
^=X®+Y"-fZ^-f 6D^YZ, ^ =^XX"-hY"-j-Z')— (l4-2/')XYZ , 
W=(X^-j-2nrZ>-f , l, = (3^^X^~(l-f2P)YZ 
)XH-.. , H,=:(3^V-(l+2/> )X-f.., 
^=C(^-]rf+z^-\-Uxyz , H -t\x^-^f^z^)-{l-^2l^)xyz 
we have identically 
TH-U1=WH,-V||,, 
and in the present case, since U=0, 
TH=WH,-V1, 
