194 MR. ROBERT RAWS ON ON SINGULAR 



Substitute this value in (66) and it becomes 



Jj-j-p + Ij-j- + VN=o. . . . (68) 



This equation, which is of the second degree in (p), is 

 expressed in terms of the differential equation (66), and 

 has in general a singular solution N=o of the envelope 

 species. 



Equation (68) includes all the particular examples given 

 by Professor Cayley and Mr. Glaisher in the papers already 

 referred to. 



17. Professor Cayley has considered the particular case 



Lp 2 -N = o (69) 



The condition of equal roots of (69) is N = o, which is 

 a solution of (69) if N is a function of y only. 



The equation L=o is a solution of (69) if L is a func- 

 tion of x only. Hence, examples 3, 4 given by Professor 

 Cayley can have no singular solutions. Their complete 

 primitives are respectively 



c + 2x±{sm~ 1 y + y \/i— y z )=o, . . (70) 



c + sin y + y */i— y z ± (sin x + x Vi— x z )=o, (71) 



which are the only solutions of these two examples. 

 Compare (69) with (57), then r + s=o and s=\/— } 



dv _ / 

 dx~V 



L 



N d^w 



dx"V L dy > (72) 



dv _ /h d */w . v 



dy~v N~daT ^ 73) 



