102 



t£ = oo, — finite, y = const., 



is a singular solution. And in the above are contained all the sin- 

 gular solutions. 



Every branch of a singular solution is either — 



A separator, only. 



A curve, every point of which has a contact of the first order at 

 least with some one real primitive, only. Or both. Or neither. 



If the first or last, it is a case of the complete primitive. And 

 such cases may be introduced at pleasure into the singular solution, 

 by writing the primitive in the form 



<p(x,y,fc)=0. 



A branch of a singular solution has at the utmost n contacts with 

 each primitive which it touches (n being determined by the nature 

 of the equation), and all of the first order, generally. Or, p x of the 

 first order, p± of the second, &c, p 1 -^2p, 2 + 3p 3 + ... being n or n — 

 (an even number) ; or some of these cases for some primitives, others 

 for others, including the possibility of some cases giving none at all, 

 when n is even. 



The branches of the singular solution which have contact with 

 ordinary primitives (whether themselves ordinary primitives or not) 

 to the exclusion of the branches which are only separators, may be 

 determined from the differential equation by the following test. 



Let ?/'= %(#,?/) be the differential equation; whence 



Find the curves which satisfy either of the following sets of con- 

 ditions : — 



Xx~ °° ^y— °° y" finite ' 



or 



p£ = oo x= const, y" finite, 

 or 



X = oo y— const, y" finite. 



Every such curve does satisfy the differential equation, and is a 

 singular solution having contact with some one primitive at every 

 point. 



And other such singular solutions there are none except those 

 designated by x=. oo, or y — oo, or both. m 



But if 



X= co and x= » f= ». 

 or 



% = oo x= const. y"= oo, 



or 



X = oo y= const. y"= oo, 



then the differential equation may or may not be satisfied ; but the 

 curve passes through the singular points of the primitives, with or 

 without contact, according as the differential equation is or is not 

 satisfied. An evolute is such a pseudo-singular solution to all the 

 involutes, passing through their cusps. 



