CONTACT AT ANT POINT OP A PLANE CUEVE. 
393 
And substituting for A, B, C their values, but attending only to the terms -which involve 
X® and Y®, the result is 
— 8^®+6«c^)X®+ • • — 8y^+6^c/’)Y®=0. 
45. But the result of the elimination must obviously be 
(X^-Y^/(Xy,-Y^J = 0, 
if (.Ti, ^ 1 , Zi) are the coordinates of the point of simple intersection. Comparing the 
two^results, and forming the analogous third equation, Ave may write 
jVi +«®— Qcag, 
=a^ -\-¥ — Sh^ -^Qabk, 
where the value of x^x^ may also be written {b-\-c~2f)[b-\-c&j~2cJ^f){b-{-cc’/ — 2ojf), 
u being an imaginary cube root of unity, and so for the other two terms. The factors 
of x^x^ might be calculated from the identical equation 
5Y^+2/YZ+cZ®=9(1 + 8Z^>=’3/V(^Y=^+2Z^+2MZ) 
-{-{(?/'+ 2 lzx)X + (;s" + 2 %)Z } X 
f ‘^xYz^ [(3/y- (1 + 2?^)^.r) Y+ (3^V- (1 + W)xy)Tl\ \ 
I — Q[(/4-2fe^)Y+(;z^+2/.r3/)Z] j 
I remark, that putting = we have 
^Y^+2/YZ+cZ^=: -ifz\ifX-^z^7.)\ 
and hence writing 1 for Y, and —1, —a7 for Z, we have 
b-\-c-2f= -tfzYf-zy, b^cai-2coY= -fz\f-u7zY, b-\-ca7-2Y= -fz\if-a?zY 
and hence the product of the three factors is —y^z^{y^—zYY—o7zY{y^—oj*zY, which is 
equal to —fz\y^—zY{y^^-zY, which vanishes in virtue of the assumed equation ^=0. 
This shows that the function 8/^ + 6 Jc/ contains the factor x. I have not veri- 
fied a posteriori, but I assume it to be true, that it contains in fact the factor .^p®, and con- 
sequently that the expressions for x,,y^, z^ are rational and integral functions of {x, ?/, z) 
of the degree 25, and containing respectively the factors x, y, z. 
46. In the theory of the cubic, a point which depends linearly upon a given point 
may be termed a derivative of such point. According to a very beautiful theorem of 
Professor Stlvestee’s, the coordinates of a derivative point are necessarily rational and 
integral functions of a square degree of the coordinates {x, y, z) of the given point ; and 
moreover, there is but one derivative point having its coordinates of any given square 
degree ird, or, as we may express it, only one derivative point of the degree 'tif. The 
successive tangentials are derivative points of the degrees 4, 16, 64, &c.; the third point 
of intersection with the cubic, of the line joining two derivative points of the degrees 
m and n respectively, is a derivative point of the degree Thus the third point 
of intersection with the cubic, of the line joining the given point Avith its second tan- 
