174 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
i.e., if u = a, v = /3 satisfy 
D [fajr] __ 
D [u, v] 
In the case where <f>, xft are each of the first degree in u, v, then the theorem 
requires to he specially interpreted, the interpretation corresponding to the fact that 
if two straight lines have two points in common, they have an infinite number of 
points in common. Hence, in this case, the two equations have an infinite number of 
solutions in common. (This particular case is of great importance in Section IV. of 
this investigation.) 
(D.) To determine the conditions that the equations cf>(u, v) = 0, xfj (u , v) = 0, may 
he satisfied by three coinciding systems of common values. 
In this case, considering as in (C) that u, v represent the coordinates of a point, the 
curves </> (u , v) = 0, xp (u, v) = 0 must have contact of the second order. 
Now 
00 dcf> dv 
du dv du 
Therefore 
d 2 cf> drcp dv d 2 cf> / di A 3 dcf> d 2 v 
du 2 du dv dv, dv 2 ) dv du 2 
d 2 v _ d 2 cf> fdcfifi ^ dcf) dcf) d 2 cp fdcfrf 
du 2 
du 2 \dv 
du dv du dv dv 3 \0% 
d±\~* 
dv, 
Hence, equating the values of d 2 v/du 2 for the two curves, there is obtained the 
further condition 
/ 0 < f >\ -3 
(fif> /0</>\ 3 2 d 2< f) dcf) d(f> d 2 cf> /dcf) 
du? [dv) du dv du dv dv 2 [du 
dv 
d^yfr /d\fr\' d 2 \fr d\fs d\fs d 2 \fr /0i|r\® / d\]r\^ 3 
_du 2 [dv ) dudv du dv dv 2 [du ) [dv 
Section II. (Arts. 2-12).—The Factors or the Discriminant, which in 
GENERAL CORRESPOND TO ENVELOPE AND SINGULAR POINT LOCI. 
Art. 2.— The Fundamental Equations. 
Let the equation of the system of surfaces be 
f(x, y , 2 , a , h) — 0 
( 3 ), 
