524 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
which consists of the inflexional tangents to a surface. It seems natural to call 
these two kinds of congruency bitangential and inflexional respectively. The fourth 
example (§§51, 52) is that of a system of conics touching six planes. The fifth 
(S§ 53-56) is that of a doubly infinite system of parabolas in one plane, the differential 
equation being a case of an extension of CLairaAuT’s form y = px + f(p), which is 
explained in §§ 53-55. 
General Theory. (S§ 2, 3.) 
§ 2. Suppose that we have n ordinary simultaneous differential equations, involving 
one independent variable x, and n dependent variables y,, y2,... Y,, with their first 
differential coetlicients p,, Po, .. - Du 
The “complete ” solution of such a system will consist of 7 equations involving 
LY ey, ANG 7 aL bibratys COUStANtS, Cine cen Cn: 
Suppose that such a solution is known. The question then arises, “ Are there any 
” 
other solutions which it does not include?” This is the question that we now seek 
to answer. 
If we take the differential equations in the form 
Si (Yas Yoo + + Yous Py +- Pr) =O (TH 1,2...n) . . . . (CL) 
and the integrals as 
Ey (ae apinne ig Cine Cn) =O (7k 2s 70) eee a GIRL) 
we have by differentiating 

oF oF, oF, 
= font r bs a eeal 5 5) re 
A, + p, on +... 4+ pu a (ORRIN Catto etree Pete Grenier (TULL, )) 
Let the system 
a OH Oe en el eee) meee ot oe So (INS) 
be an integral of (I.). 
From (LV.) we derive 
On, A, 
Set Pigg ts. + Peg = 0 (== 1p?) 327) Sow eee Ae 
a 1 
By eliminating p,,...p, from (III.) and (V.) we find such values of ¢, ¢g,... Cy, as 
will make (111.) and (V.) equivalent* ; but if the values of p,,...p, given by (III.) 
* This argument must be somewhat modified if the equations (IJI.) are not enough to give the 
values of ¢, ¢,... Ca. Suppose that m independent equations, and no more, can be formed from (IIL.) 
not containing the quantities ¢,...¢x. These equations must be included in the system (1.), and are 
therefore equally satisfied (possibly in virtue of (IV.)) by the values of p,,... pn given by (V.). 
The other n — m equations of the system (III.) may then be transformed by substitution of the values 
