564 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
Put now s =n, 7 = 0, and we have finally 
n(n — 1 : 
(= Pal 7S Maya SS Gy = © Gand? oP ae Un2 B... + ( — 1)"a, a, 
the constants being connected by the relation 
O((Ghy Chy oo» Ge) = O, 
§ 54. The factor ' : : 
Di Te a Oe yee) 
ee ee ste Gn ” 
equated to zero, leads to the singular solutions, the first integral being found by 
elimination of p, from ¢@ = 0 by means of it. 
Thus the solution of $(q, q.--) =O is exactly on the lines of that of 
Ws (ap; — ¥, P,) = 0, which is CLatravutT’s form. 
§ 55. The equations to be integrated in finding the singular solutions are 
a 


da, diy __ dts Ut, 
ity tht Ohi tng 
(hy Chis a 6» Gn) = 0, 
For example, when 7 = 2, we have 
9 
da,” = day das, 
@ (Ao, %, A) = 0. 
If (a, @, @) ave taken as Cartesian coordinates, the solution represents curves on 
the surface ¢ = 0, the tangents to which are parallel to generators of the cone 
@,° = Mt, that is to say, meet a certain curve at infinity. The second singular 
solution is given by forming the envelope of such curves, which does not generally 
exist, but may in particular cases. 
Example VI. (§ 56.) 
§ 56. As an example take the equation 
(2yp, — py’)? = 4p2 (Pi — xp2)*, Or (Yo2 — GV)’ + 4qom? = 0, 7 beg 2. 
The ccmplete primitive is 
dacky = 1+ 2c + daca + Adateea®. 
The equations giving singular solutions are 
(1 + 6e?ex). da + 6ax.de = 0, 
344 
; 2¢ C 
(a = Sa | da = mee de a) 
