184 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
where more than one system of distinct values of a, b exist, and ( 2 ) the particular 
case of the preceding in which two systems of values of a, b exist which coincide.] 
In this case equations ( 11 ), (25), (26) are satisfied. 
Hence when x = £, y — 77 , 2 = £, 
A = 0 
and 
a a 
dx 
becomes 
* = f 
y = v 
Hence when x = £, y = y, z = £, 
3 A/g/ _ 3A/p/_ 3A/!)/ 
0.r / Dx dy / Dy dz J Dz • • • • • \ 
Hence the tangent planes to A = 0 ,f(x, y, 2 , a, /3) . = 0 , coincide at £ y, £. This 
proves that the locus of ultimate intersections is generally an envelope. 
It should be noticed that the proof shows that the locus of ultimate intersections 
touches in general at each point on it one of the infinite number of surfaces of the 
system passing through that point. This will be referred to in future, to distinguish 
it from more complicated cases, as a case of the ordinary envelope. 
It should also be observed that the above conclusion cannot be drawn if 
x = £j, y = y, z = £ make D//D.r = 0, DfjJdy = 0, Df/Dz = 0. 
Hence the investigation itself suggests the examination of the case in which a locus 
of singular points exists. 
Art. 9.— To prove that if E = 0 be the equation of the Envelope Locus, A contains 
E as a factor once and once only in general. 
(A). If x = £ y = y, z = £ be a point on the envelope locus, then suppose that 
the values of a, b satisfying (11), (25), (26), are a, 
Then one of the systems of values of a, b satisfying (4) and (5), must become equal 
to a, /3 when x = y = y, z = £. 
Suppose that a x becomes a, b x becomes /3. 
Hence A becomes Kf(^,y,C,«-,/3), where IT is what B, becomes, and therefore 
A vanishes. 
Hence by Art. 1, Preliminary Theorem B, A contains E as a factor. 
Further, A does not contain E more than once as a factor in general, for the value 
of 3 A/dx given by (53) does not in general vanish. But it would vanish if A con¬ 
tained a power of E above the first as a factor: for suppose A = E“. i|/, where m is a 
positive integer greater than unity, and \p some rational integral function of x, y, 2 . 
