[118] PROFESSOR DE MORGAN ON SOME 



, dx dy 



T 'da ' " da 

 - dx , dy db' 



f« + f*^-+f ! ,T L + ^r= °> 



da da da 



and similar equations with respect to ft. Determine \ and ft. from \<p x + fi\^ x + f x = 



dft' dft' 



— = - \<£„- fj.yj/ a -f a , -rr- = - \(j>b - H^b- fb- 



The singular solution usually takes place when X and n are infinite ; or when 



*•*-**-* o t * (£-jj) f -* (^-f:),- «■ 



Common differentiation shews that 



This reduces the preceding condition to 



*■©-* a— ©.-©/-°- 



Now resume the equation y" + F y .y + F x <=> 0, and let us ask what is the condition under 

 which y" is the same both for the primitives and the singular, so that their contacts are of 

 the third order. This happens when 



(F ay + F hy . ft') y' + (F ax + F bx . 6') = 0, 

 where y + F m 0, F a + F b . ft' = 0. Substitution shews that we have in this case also 



©.-(!)/- 



So that if ft' = x (a, 6) be derived by eliminating x and y between 



0=0, a+ 0».ft'=O, it*) +(%) .ft'= ; 



the ordinary primitive values of ft in terms of a, substituted in d> (x, y, a, ft) = 0, give 

 families which have contacts of the second order at least with their singulars. But a singular 

 value of 6 in terms of a, similarly used, gives contacts of the third order. 



The connexion of the preceding with the theory of the singular solutions of equations of 

 the second order will be noticed, and also the glimpse which it gives of the general theory 

 of singular solutions. I will conclude this section with a remark which may tend to increase 

 confidence in that main-survey system of reasoning which Lagrange and others have used, 

 though by no means intended to imply that it is to be implicitly received. 



An equation of the form d> (,r, y, c) = may have many different forms given to it. 

 If \1/ (x, y, c) = be another equivalent equation, we know that (p c : <p x = \f/ c : yf/ x . Now it 



