AND LTNES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
173 
Therefore they make p — 0. 
Therefore p contains (/> as a factor. 
Therefore \Jj contains <f> m + 1 as a factor. 
Hence, if the theorem is true for a special value of m, it is true for the next value. 
But it has been proved true when m = 2, hence it is true in general. 
(C.) If u, v be determined as functions of other quantities by the equations 
(f)(u, v ) = 0, i fj(u, v) = 0, 
ivhere 6 and ifi are rational integrcd functions of u, v and the other quantities; then, 
if two systems of common values of u, v become equal, they will also satisfy the equation 
D [<fr> -f] 
D \_u, v ] 
= 0 . 
Conversely, if values of u, v can be found to satisfy at the same time the three 
equations 
(f) (u, v) = 0, xfj ( u, v) = 0, 
B [0. _ ( ) 
D [u, v] 
then these values count tivice over among the common solutions of the equations 
<f) (w, v) — 0, xjj (u, v) ■= 0, 
except in the case ivhere (f> and xJj are of the first degree in u and v; and then the two 
equations have an infinite number of solutions in common. 
To prove this, let u, v represent the coordinates of a point in a plane. Then 
(f> (u, v) = 0, xJj(u, v) = 0 are the equations of two algebraic curves. 
The values of u, v which satisfy at the same time both equations are the coordinates 
of the points of intersection of the two curves. 
Let u = a, v = f3 be the coordinates of one point of intersection. The tangents to 
the curves at a, /3 are 
/TJ \ ( (*’ ft) I /-XT p\ ) Ci ’ ft) A 
(U - a) + (V _ /3) d ±^ll = o, 
where U, V are current coordinates. 
The two tangents will coincide, i.e., the curves have two coincident points of 
intersection, if 
0<£ («, /3) 3-v/r (a, /3) d(f) (a, yd) d\fr (cc, f3) 
