142 
PROFESSOR M, J. M. FULL ON THE LOOPS OF SINGULAR POINTS 
PART I.—THE EQUATION OF THE SYSTEM OF SURFACES IS A RATIONAL INTEGRAL 
FUNCTION OF THE COORDINATES AND ONE ARBITRARY PARAMETER, 
Section I. (Arts. 1-6 ).—The factors of the Discriminant which in general 
CORRESPOND TO ENVELOPE AND SINGULAR LlNE LOCI. 
Art. I. — 2b show that if E = 0 he the equation of the Envelope Locus, the Discrimi¬ 
nant contains E as a factor. 
Let the equation be 
f{x, y, z, a) = 0.. (1), 
where x, y, z ai'e the coordinates, a the parameter, and f is supposed to be a rational 
integral function of x, y, z, a. 
Denoting partial differentiation when x, y, z, a are treated as independent variables 
by D, the locus of ultimate intersections can be obtained by eliminating a between 
(1) and 
D/(r,?/, z,a) _ . 
La . 
Let the roots of (2) treated as an equation in a be cq, a 2 , . . . , which will at first 
be supposed to be all different, so that they do not make 
D 3 / ( x > y,z,a) _ 
W ~ 
Then if K be a factor introduced to make the discriminant A, obtained by 
eliminating a between (1) and (2) of the proper order and weight, 
A = 11/ (x, y, z, a 1 )f(x, y, z, a z ) . . . (3). 
Let x — f y — y, z = £ satisfy (!) and (2) when a = a. 
Suppose that cq becomes a, when x = y — y, z = £, 
therefore, 
/(£ i?, i, °0 = o ..(4), 
- d! =°.F»- 
Now in A put x = f y — y, £ = £; and consequently oq = a, therefore f ( x , y, z, cq) 
becomes /(f, y, £, a) and consequently vanishes. 
Therefore A vanishes when x = / y = y, z = 
The next step is to show that the locus of ultimate intersections is the envelope. 
Write, for brevity, 
A = Q f(z, y, z, cq) = Qf .(6). 
