530 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
Now, in general, the coefficient of B on the left does not vanish when 
O=0=F,=...=F,, and therefore B must vanish for such values, and must be 
of the form 
C.F, + C.F, +...+ Ca. 
Thus with a shght change in the meaning of A,, A... .., 
E=Co?+ A\F, +...+A,F,. 
Now, AB — 0:— AK, = 1.) identically, “and! 0/— 0) AQ) —10; i Am 0) —s0mor 
systems of values that satisfy one of the 7" system of singular solutions. Hence in 
such a case 
B= 0, N= O0,.5 Vs = ©, 
and the number of consecutive solutions of the equations 
=O, Si... , 674 1. 
Geometrical Interpretations. (S§ 10--13.) 
§ 10. The geometrical application of the above theory to curves in space of n + 1 
dimensions is easy. 
The equations (II.) may be taken to represent a series of curves (that is, singly 
infinite continuous series of points) in such space. Through any point a certain 
number, z, of such curves may be drawn, 7 being the number of solutions of (II.) 
when solved for ¢), Cy, ... Cy 
At every point such a curve is met by 2 — 1 other curves of the system, and at 
certain points one of these 7 — 1 curves coincides with it. 
The direction of the tangent to such a curve is given by the equation (I.) or (III). 
The first smgular solution is the envelope of a series of curves of the system, each 
of which meets the consecutive one, and the locus of all such envelopes is the surface 
(7 infinite series of points) whose equation is E = 0 (see § 4). 
In general, the 7" singular solution is the envelope of a series of curves belonging 
to the (7 — 1)" singular system, and such that each meets the consecutive one; the 
locus of such envelopes is an (7 — + -+ 1)?” infinite continuous series of points at 
each of which 7 + 1 consecutive curves of the original system meet, as also do 
r—s-+1 of the s™ singular system. All the successive singular curves touch the 
original curve. 
§ 11. If every curve of the first system (II.) has a node, the equation (VII.) will be 
satisfied at every point of the node locus, but (VIII) will not (compare § 4). If 
