Equations of the First Order, 359 



tical effect of this connexion being to produce the final result in 

 the form of an equation in c of the same degree as the differen- 

 tial equation is in p. With regard to the latter equation, I 

 assume that its coefficients are rational, or rather unambiguous 

 in value. Should these coefficients contain radicals or ambi- 

 guous forms, they must be expelled before we can determine the 

 real degree of the equation in p. "We cannot, however, assume 

 that the coefficients of the complete primitive in c will also be 

 rational, the process of integration being apt to produce trans- 

 cendental forms in lieu of rational algebraic ones. 



The investigation which follows will, I think, be found to solve 

 this problem completely for equations of the second degree; and 

 it exhibits some processes of solution which possess an indepen- 

 dent interest. 



Proposed Theory. 



5. Having indicated the object in view, I now proceed to the 

 consideration of differential equations of the first order, begin- 

 ning with those of the second degree. These equations are of 

 the form 



^_2(/>^ + 2 =O, 



(f> x and <p 2 being unambiguous functions of x and y. 

 It is obvious that in general a primitive of the form 



c 2 - 2^+^ = 



will give, by the elimination of c, a differential equation of the 

 form above written, in which (/> x and <£ 2 are easily obtained from 

 i/r 1 and yjr 2 . This proposition, however, although true in general, 

 admits of a particular exception. If yfr 2 happen to be a function 

 of yjr l (say/v/rj), we merely obtain by differentiation 



' &=/, 



so that c does not contain/*; and the requisite elimination is 

 not possible. 



The primitive may be placed conveniently in the form 



(cf)*-2(cf)++i; ' 



which gives for c the two values 



and this (remembering that c is arbitrary, and that therefore any 

 function of c may be written for c) admits of being placed in the 

 form 



C = /Lt + ^ 



•xlr 



in an infinite variety of ways. Thus we may make fi= r f , and 



