534 The Eev. R. Carmichael on the Singular Solutions 



indicating any of the characteristics by which such equations as 

 do not admit of its application may be a priori recognized. 



Light seems to be thrown upon the subject by the circum- 

 stance that, as is readily seen, the equations last written are, 

 respectively, the conditions that the previous equations to which 

 they belong should have equal roots, the former being regarded 

 as a quadratic in j9, the latter in yp, the result in neither case 

 being reducible to the Clairaut type. Again, in the equation 

 numbered (8) in the same collection, 



[xp —y)[xp— 2y) -\-a^=0, 

 or 



[xp — yf — {xp — ij)y + ^ = 0, 



the condition for equal roots in [xp—y) is 



and the result obtained by the theory is a true singular solution, 

 inasmuch as it satisfies the given equation. It appears, too, 

 that if P be the perpendicular upon the tangent at any point 

 of this parabolic curve, and « the angle which it makes with the 

 axis of X, the following relation always holds, 



P'^ + Py sin u-\-x^ sin^ « = 0. 



The remark previously made will perhaps be more readily ap- 

 prehended by observing that the symmetrical form of the above 

 equation is 



/ du dii\^ ( du du\ du .,fdu\^ ^ 

 \^Tx+^Ty) ■^VJx^yTyJyTy^^Kdy) =^' 

 which becomes in the case stated, 



/ du du\ , ydu du %du 

 VTx^yTy)-^2Ty=^Tx^y dy^.=^' 



an equation of the linear or Clairaut type, and the solution of 

 which is 



U=iUq[x, j/*)=0, 



where Mq is any homogeneous function of the order 0. 



The necessity of clearing any proposed equation of radicals 

 and fractions is easily intelligible, from the consideration that 

 any equation linear with respect to p, but which involves radicals 

 or fractions, can only be regarded as a factor of some equation 

 of higher degree. In equations reducible to the Clairaut type, it 

 is evident from the examples previously given that such a prelimi- 

 nary operation is unnecessary. As regards the alleged necessity 

 of verification by substitution of any supposed singular solution in 

 the given equation, the following example will suffice. If it had 



