100 
1874, vol. XXXVI., p. 179). The Rev. Robert Carmichael 
has given a paper on Singular Solutions (Phil. Mag., 1858, 
vol, XV., p. 522), on which I need not here comment. 
13. Professor Cayley has also (see Messenger of Math, 
for May, 1872, p. 6; for June, 1876, p. 23), given a new 
theory of the whole subject which may be stated, not per- 
haps with perfect fulness, as follows : The locus represented 
by the discriminant, with respect to the arbitrary constant, 
of the primitive is made up of the nodal locus twice, the 
cuspidal locus three times, and the envelope locus once; 
and the locus represented by the discriminant with respect 
to p of the differential equation is made up of the envelope 
locus, cuspidal locus and tac-locus, each of them once. 
14. This theory of Professor Cayley’s seems to account 
thoroughly for the extraneous factors which occur in the 
processes for obtaining singular solutions. 
14. Take a set of circles each of which touches both axes 
of rectangular coordinates and apply the Cayley theory. 
The primitive is (y - c) 2 + (x- c ) 2 = c 2 , the c-discriminant is 
- 2xy and xy = 0 is made up of the envelope-locus once, viz., 
x = 0 appears once and y = 0 appears once. The ^-discrimi- 
nant is - 2 xy(x - y ) 2 , which is made up of the envelope-locus 
once and of the tac-locus. At first sight the tac-locus seems 
to occur twice. But x - y is the bisector of the right angle 
at which the axes meet, and there is external contact of 
non-consecutive primitives at each end of that diameter of 
any one of them which when produced passes through the 
origin ; and I imagine that when the parameter gradually 
varies each end of such a diameter must be held to trace a 
distinct tac-locus, so that (x-yY = 0 gives each tac-locus 
only once. If this be so then, generally, when two non- 
consecutive primitives touch a primitive not consecutive 
to either of them each point of contact must be held to give 
rise to a distinct tac-locus, even when the path taken by 
one such point is the same as that taken by the other. 
2, Sandringham Gardens, Ealing, 
Near London, W., March 16th, 1882. 
