OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 531 
every curve has a cusp (VII) and (VIII.) will be satishied by the equation to the 
cusp locus, but it will not fulfil the condition of being a first integral. If different 
curves of the system touch, the tac-locus will satisfy the equation (VIII) but not 
(VIL). All this is exactly parallel to the known theory of single differential equations 
of the first order. 
§ 12. There is also an application to plane curves. 
Suppose that the differential equations (I.) include the following :— 
1D) = Up joy = Gey 3 ona Ui 
Then the system (I.) simply reduces to an equation of the 1 order and its n first 
integrals make up the system (II.) from which the singular solutions are derived. 
If we write the differential equation 
F(%; Ys Pr Pz +++ Pn) = 0, 
putting y for y,, then the equation (VIIT.) becomes 
of / Op, = 0, 
and this is the first integral (if it is one) from which the singular solutions are 
derived. 
The final integral, found by eliminating p,, py... Pa—, from the system (II.), is the 
equation to a system of curves, of which there are 7 passing through any point and 
having at that point contact of the (n — 1) 
it ; all these have contact with one another of the (7 — 1)" order. 
" order with any assigned curve through 
If two of them coincide, then a curve of the first singular system passes through 
the point and has contact of the (7 — 1)" order with each of the 7 curves, and of the 
n™ order with either of the two coincident ones. 
A curve of the first singular system can be made to have contact of the (7 — 2)™ 
order with any given curve at any point of it. 
At any point of a curve of the second singular system three coincident curves of 
the original system and two of the first singular system will satisfy the conditions 
for contact with it of the orders n — 1, n — 2, respectively, and in each case the 
contact will be actually of the 7™ order, and so in general. The single curve of the 
n™ singular system is the envelope of those of the (mn — 1)", but it is more than 
an ordinary envelope since its contact with each of the enveloped curves is of the 
nu order. 
§ 13. It should be noticed that ifm — 1 of the dependent variables are eliminated 
by differentiation from 1 simultaneous equations, the new equation, of order n, will be 
satisfied by the same complete primitive, but that its singular solutions will generally 
3Y 2 
