SOLUTIONS OF DIFFERENTIAL EQUATIONS. 173 



The same process is repeated in the examples 2, 3, 4, 

 and it is scarcely necessary to remind mathematicians of 

 the close proximity of this process to the two theorems 

 (A) and (B). 



Again, in the c Messenger of Mathematics ' for May 

 1882, J. W. L. Glaisher, M.A., F.R.S., has an interesting 

 paper, including pages 1 to 13, entitled " Examples illus- 

 trative of Professor Cayley's Theory of Singular Solution." 

 The theorems (A) and (B), or the condition of equal roots, 

 are applied by Mr. Glaisher to the solution of twenty-one 

 examples, and he states that " in my lectures on differ- 

 ential equations I have for some years illustrated Cayley's 

 theory of singular solutions, of which the preceding 

 paragraphs contain a short resume, by several simple 

 examples." 



As, however, neither Professor Cayley nor Mr. Glaisher 

 has given a proof of the two theorems in question, I will 

 endeavour to supply a demonstration of them, and to show 

 the range of their application to determine the singular 

 solution both from the complete primitive and its derived 

 differential equation. 



2. It is well known that the most general differential 

 equation of the first order and of the nth degree is 



where the roots with respect to f -p) are r x> r z . . . r n 



functions of oc, y. 



The complete primitive of (1), therefore, is an equation 

 of the form 



(c x + R f ) (c x + RO (c n + R n ) = 0, . . (2) 



