SOLUTIONS OF DIFFERENTIAL EQUATIONS. 197 



Mr. Glaisher states the condition of equal roots of (77) 

 to be 1— x z . 1— y z = o. All I can say at present is that 

 1— x z = o satisfies the condition of equal roots in a peculiar 

 way. 



It may be stated that (jj) can be replaced by its 

 equivalent 



The roots of which, with respect to I -7- ), are equal when 



w=+i, a condition which satisfies (82). This principle 

 can be applied to equation (65), with a view to show the 

 cases in which L=o may produce a singular solution as 

 well as N=o. 



18. Professor Cayley states that if Lp 2 + 2Mp + N=o 

 has a singular solution, it is either M z — LN = o or a 

 factor of M a —LN equal zero. To this proposition may 

 be added another, viz. 



has a singular solution M = o, which does not appear 

 to be included by the above proposition of Professor 

 Cayley's. 



