POINTS OF THE INTEGRAL CALCULUS. [117] 



first differential equation is obtained, both for primitive and singular, by eliminating a, after 

 substitution for b, between 



$ = 0, y + F = 0, where (F - <p x + <p v ). 



Now let a be the function of x and y which gives the singular solution. We have then, 

 for the singular solution, 



y" + F, + F y .y'+ (F a + F b . b') (a x + a y .y') = 0, 



whence, in order that the singular solution may agree with the primitives which touch it, as 

 to y", and so have contact of the second order with them, we must have 



F a + F b .b'=0. 



If then we eliminate x and y from 



<p=0, (<p a +<l> b l>' = or 0,- os , d) y = co), and F a + F b . 6'= 0, 



we have a differential equation of the first order between a, b, b', each solution of which, be 

 it primitive or singular, selects a family of curves out of <p = for which there is contact of 

 at least the second order with the singular curve. 



Take for examination (p a + (p b .b' = 0. We have then <p a F b -(p b F a = 0, which, combined 

 with (f> = 0, gives x and y in terms of a and b. Take the pair cf> = 0, y + F = 0, and by 

 help of the relation between b and a, suppose a a function of x and y from the first, which, 

 on substitution, gives ^ for y in the second. We have then 



0,+ (0„+0 4 .6')^-o, i&+J?, + (F a +F t .b')~- o, &c 



<*X F , jkt**lg rf, d X F , ^H-^-ft' ■ 



rfy~ y ipa+^.b'- 9 ^ dx ^"V+^.6'-^- 



Whence, as a general rule, by passing over exceptional cases, and leaning to the conclusion 



that - will appear in specific cases when it appears in general forms, we should imagine that 



v, and Xj, would be infinite when the singular solution has contact of the first order only 



with primitives, and - when of the second and higher orders. But such instances as I 



happen to have chosen do not confirm this conclusion. 



Let us now take a singular solution of the equation between a, b, b'. Taking the three 

 equations 



T <pb F b 



and calling the second \// = 0, and the third /+ b' = 0, we imply in them the function which 

 b' is of a and b in the differential equation whose singular solution we are in search of. And 

 we have (considering x, y, b', as functions of a and b), 



dx dy 



