30 KEPORT — 1875. 



has no general integral, it will only happen exceptionally that it will have a sin- 

 gular solution. When this singular solution exists, we have simultaneously the 

 three relations 



/=»' !=»' S + l^-O (A) 



or 



*=0, |,=0, !*.■ + !* =0 (B, 



for the values of x and y which satisfy the singular solution *, 

 (II.) If the equation (1) has a general integral, 



^{F,y,c) = ^\ (2) 



the singular solutions are given by the elimination of c between the relations 



P = ". I = »] (3) 



or 



p=^' 7:=^]' w 



or by the systems 



[/=»' |-»] « 



or 



equivalent to (3) and (4), unless we have 



^0=^--=^' -$=0-* (7) 



Besides, in general, the two equations (5) or (6) have as a consequence the third 

 equation (A) or (B), contrary to what takes place in the first case t. 



2. M. Darboux, to whom is due the subdivision of the subject as indicated above, 

 has given several examples where the rule II. seems to fail. We shall show that 

 this is not the case if we introduce in the injinitedmal analysis the notion of singular 

 solutions situated wholly at infinity. 



(I.) The differential equation 



has for its general integral 



(y_c)2_a;3 = 0, Qvx—(y-cf=0. 



The system (5) cannot give a singular solution. The system (6) leads to x=0, 

 which does not satisfy the equation, as it belongs to the case of exception (7), viz. 



J— 5-= 00 ; the system (6) is not equivalent to (4). The latter gives 



dx _ —2 _^ 2 _f. 

 d'c ~ 3(y-c)^ ~ 3Va- ~ ' 



viz. x= cp. Now a; = 00 is really a tangent to the cubic {y-c)'—x^ = 0, as is im- 

 mediately evident. 



* Darboux, 'Comptes EenduB,' t. Ixs. pp. 1329-1383 ; Catalan, 'Comptes Eendus,' 

 t. Ixxi. pp. 50-57 ; Darboux, ibid. pp. 267-270 ; 'Bulletin des Sciences Math^matiques et 

 Astronomiques,' t. iv. pp. 158-176. 



+ P. Mansion, 'Bulletin da Bruxelles' (2), t. xxxiv. pp. 149-167; Gilbert, ibid. 

 pp. 142-145; P. Mansion, 'Bullettino de Boncompagni,' t, vi. pp. 283-285 (Luglio, 1873). 



