172 Mil. ROBERT RAWSON ON SINGULAR 



differential equation, are unnoticed by Boole, Hymcr, De 

 Morgan, Abbe Moigno, and others. 



It seems, therefore, only fair to conclude that the above 

 distinguished writers on the solution of differential equa- 

 tions failed to see or perceive the theorems (A) and (B) in 

 the incidental remarks made by Lagrange, and referred to 

 by Sir James Cockle in paragraphs 9, 10, n. 



Had Lagrange himself perceived the force and gene- 

 rality of his incidental remark, no doubt he would have 

 deduced the theorems (A) and (B), and would have applied 

 them to the solutions of the numerous examples which are 

 given in the 'Lecons sur le Calcul des Fonctions/ as 



, . „ .. . , dy dx dp 



being more easy or application than are -~ , -=- t -j-, 



The two theorems (A) and (B) were communicated to 

 my mathematical friends Sir James Cockle, Rev. Robert 

 Harley, Mr. Woolhouse, Mr. Purkiss, who was senior 

 wrangler in 1864, Dr. Hirst, and others many years ago. 

 It is now upwards of twenty years since these theorems 

 were explained and taught to my pupils in Her Majesty's 

 Dockyard Schools, Portsmouth. 



It seems to be necessary to enter somewhat into these 

 particulars, as Professor Cayley, of Cambridge, has com- 

 municated two very comprehensive papers " On the Theory 

 of the Singular Solutions of Differential Equations of the 

 First Order'' to the ' Messenger of Mathematics/ vol. ii. 

 1872, p. 6, and vol, vi. 1876, p. 23. In these papers 

 Professor Cayley has remarked " that the discriminant of 

 the differential equation in regard to (p), and that of the 

 integral equation in regard to (c), are each equal (y z — 1), 

 and we have a true singular solution (y 2 — 1)" Ex. 1, 

 p. 24. 



