376 Prof. Cayley on Differential Equations and Umbilici. 



that the conclusion may be extended to singularities of a higher 

 nature, viz. the factor corresponding to any singular curve which 

 presents itself as part of the complete envelope divides out from 

 the derived equation ; and such singular curve does not consti- 

 tute a solution of the reduced equation, but we have a singular 

 solution corresponding to the reduced or proper envelope. 



II. 



Consider the differential equation 



y(p 2 — 1) + 2mxp = 0, 



where, to fix the ideas, m > or = 1 ; the integral equation may 

 be taken to be 



z = (mx + ^SMf) [mx> + y* + VwV+?) m " l ', 



or rather, writing for shortness □ =m 2 ^ 2 + z/ 2 , and putting 



r=(w + V / D)(m^ + ^ + ^v / Df- 1 = P + QV / D ) 



the integral equation is 



(*-P) 2 -Q 2 D =0, or * 2 -2P,z + P 2 -Q 2 a =0, 

 where 



P 2 -Q 2 □ = (mV- □) {(^ 2 + 2/ 2 ) 2 -^ 2 D \ m ~ l 



= —y* m {y' 2 + 2m—lx' 2 \ m -\ 



In the particular case m = 1 the equation is 



z = x-\- ^a? + y*, or z* — 2zx—y' l — 0. 



Before going further, I remark that, m being a positive integer 

 greater than unity, we have 



*= P + Q\/n = mx(mx* + y*) m ~ l 



+ {ma* + y 2 + m^lma?) (mx* + y 9 ) m " V D + &c -' 



the subsequent terms being divisible, the rational ones by □ , and 



the irrational ones by Dv D. Hence, observing that 



mx 2 + V* + [m -l)?nx <2 =m <2 x~ + y 2 = Q, 



we see that Q contains the factor Q, and the equation D =0 

 belongs to a cuspidal curve on the surface. If however m=l, 



then the equation is s = a? + v D, so that Q =1 does not con- 

 tain the factor D; and D =# 2 + 2/ 2 =0 is not a singular curve on 

 the surface, but is in fact the reduced or proper envelope. 



The curve represented by the integral equation will pass 

 through the origin (#=0, y = 0) for the value z = of the con- 



