806 PROFESSOR CHRYSTAL ON THE 



In general (10) will have two distinct roots ; and we conclude that in general, ivhen 

 the primitive touches the ^-discriminant tivo integral curves touch each other there, and 

 ive have a tac-point. 



If a branch of the integral curve touch the ^-discriminant at every point of the 

 latter, and not merely at a single point, as hitherto supposed, then the p-discriminant 

 is a solution of (7), and (6) is a second approximation to a solution of (7). We must 

 then have 



4aJc i +(2\+e )Jc+d =0 (11). 



In other words, k is a root of (10). Hence one of the solutions furnished by (10) is the 

 j>discriminant itself. The other, provided it be distinct, is a primitive touching the 

 p-discriminant. 



The condition (11) may be written (since 4a 2 ±0>) 



Now 



Za 4«„ 4a„ 



h 2 — 4r/ rl 

 h\ f J — u i ^¥'0 1 ,7 • 



4« 2 



Hence (11) may be written 

 that is to say, 



This last condition reduces k to 

 hence the other root of (10) is 



4a 2 



(/v + V4a 2 ) 2 = 0, 



\=-d /b. 

 The approximations to the ^-discriminant and the primitive, therefore, are 



2?-discriminant, y = — j±- or? n$y 



Primitive, y=—f±x i / 14 ^ 



which are in general distinct. Hence, if any branch of the ^-discriminant is a solution 

 of the differential equation that branch is an envelope singular solution ; that is to say, 

 at every point there is a primitive distinct from the branch of the ^-discriminant in 

 question which touches that branch. 



