202 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Section III. (Arts. 13-15).—Consideration of the cases reserved in the 
PREVIOUS SECTION, IN WHICH TWO SYSTEMS OF VALUES OF THE PARAMETERS 
SATISFYING THE EQUATIONS, D f/Da = 0, Df/Db = 0, COINCIDE AT A POINT ON 
the Locus of Ultimate Intersections. 
The interpretation of the condition 
[a, oc] [(3, (3] — [a, /3] 2 = 0, 
which is marked (76), when the equation of the system of surfaces is of the second 
degree in the parameters, is different from its interpretation when it is of a higher 
degree. 
It will be supposed, in this section, that the degree of the equation of the. system 
of surfaces in the parameters is higher than the second. 
Art. 13.— To prove that if each Surface of the System have Stationary Contact with 
the Envelope, then A contains E 3 as a factor. 
(A.) It will be shown that when the condition (76) holds in the case of an envelope 
locus, the curve of intersection of the envelope with each surface of the system has a 
double point at tlm point of contact, such that the two tangents coincide. [Such 
contact is called stationary (see Salmon’s ‘ Geometry of Three Dimensions,’ 3rd 
Edition, Arts. 204, 300).] 
To prove this, it is necessary to find the direction of the tangents to the curve of 
intersection of the envelope and one of the surfaces of the system at the point of 
contact. 
Let £ y, £, be a point on the envelope. Let the surface touching the envelope at 
this point be 
f(x, y, z , a, (3) — 0, 
which has been marked (9). 
Then equations (11), (25), (26) hold. 
Let £ + S£ y + S17, £ + be a point near to 77, £, which lies on the curve of 
intersection of the surface (9) and the envelope. 
Since it is on the envelope, it will be the point of contact of one of the surfaces of 
the system. 
Suppose it is the point of contact of the surface (10). 
Then 
f f + §£ V + S-*7» £ + a, /3) = 0.(97), 
and the equations obtained from (11), (25), (26) by changing £, 77 , £, a, [3 into £+ 
y + &y> £ + S£, a + Sa, (3 + S/3. 
