SOLUTIONS OF DIFFERENTIAL EQUATIONS. 177 



Differentiate (10) with respect to (c), reject the common 

 factors ; then 



dy ±2 \/w s \ \ dy ±2 \/w z 



dc , — dv T dw t \ \ dc , — dv, dw 



dy dy ) \ ~ dy dy j 



.... &c. = o. . . . (13) 



Now the condition of equal roots gives tv 1 = o, w z = o, &c, 



du 

 therefore -~ in (13) is evidently zero. It may be shown 



dx 

 in a similar manner that -=- is zero also. 



dc 



In the case of the envelope species of singular solutions 



dy dx 



■*- == o and ^- = are equivalent. Boole states that such, 



" although not necessarily equivalent, do not lead to 

 conflicting results" (Boole's DifF. Eqs. p. 146, 2nd ed.). 



For a particular hypothesis with respect to the form of 

 the complete primitive, viz. 



j"Q(c-X) fl Vfc-y=o, .... (14) 



where Q is a function of x, c, which neither vanishes nor 

 becomes infinite when c=X, Boole has proved that for a 



singular solution f -J- J = infinity. Boole also states "that 



inquiries which are scarcely of a sufficiently elementary 

 character to find a place in this work indicate (with very 

 high probability) that this character is universal and inde- 

 pendent of any particular hypothesis, and that it consti- 

 tutes a criterion for distinguishing solutions of the envelope 

 species from others" (Boole's DifF. Eqs. p. 163, 2nd ed.). 



The proposition, viz. -—=■ infinity, which seems to have 



originated with Laplace, and was subsequently investigated 

 by Lagrange, Cauchy, Poisson (see Boole's Diff. Eqs. pp. 

 172, 174, 2nd ed.), may be proved generally as follows : — 



