AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 27 
Part II.— The equation of the System of Surfaces is a rational integral function of the 
COORDINATES AND TWO ARBITRARY PARAMETERS. 
Section I. {Art. 1 ).—Preliminary Theorems. 
Art. 
1.—(A.) If £ 7 , £ are the coordinates of any point on the locus 0 {x , y, z) = 0 (where 0 is a 
rational integral indecomposable function of x, y, z ), and if the substitutions x = f, 
y = r]' z = f make 0 (x, y, z) and all its partial differential coefficients with regal’d 
to x, y, z up to the tt th order vanish, and if they also make any one of the partial 
differential coefficients of the {n + l) th order vanish, they will also make all the 
partial differential coefficients of the {n + l) tl1 order vanish (0 being a rational 
integral function of x, y , z, but not in general indecomposable). 
(b.) (i.) if r, 7 , £" are the coordinates of any point on the locus 0 {x, y, z) = 0 (where 
0 is a rational integral function of x, y, z which contains no repeated factors), and if 
the substitutions x = D y = 7 , z = f make 0 C x > y, z) = 0 (where 0 is a rational 
integral function of x, y, z), then 0 contains the first power of 0 as a factor. 
00 
(ii.) If x = g, y = 7 , a = make 0 = 0, gy = 0, then 0 contains the second power 
of 0 as a factor. 
00 0" i— l0 
(iii.) If a; = y — 7 , z = ^ make 0 = 0, -g— = 0, . . . g-—=0, then 0 contains 
0 "‘ as a factor. 
(C.) If m, v be determined as functions of other quantities by the equations 
0 {u, v) = 0, 0 (m, -r) = 0, 
where 0 and 0 are rational integral 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 [ 0 . 0 ] 
D \u, v ] 
0 . 
Conversely, if values of u, v can be found to satisfy at the same time the three 
equations 
0 {u, r) = 0, 0 (m, v ) = 0, D[ti, v ] = 
then these values count twice over among the common solutions of the equations 
0 («, v) =0, 0 (m, r) = 0, 
except in the case where 0 and 0 are of the first degree in u and v ; and then the 
two equations have an infinite number of solutions in common. 
(D.) To determine the conditions that the equations 0 (u, v) = 0, 0 ( u , v) — 0 may be 
satisfied by three coinciding systems of common values. 
PAGE 
169 
171 
173 
174 
Section II. {Arts. 2-12 ).—The factors of the Discriminant , which in general correspond to Envelope and 
Singular Point Loci. 
2. —The Fundamental Equations. 174 
3. —The Loci of Singular Points of the System of Sui’faces. 175 
Example 1. 176 
