Equations of the First Order. 371 



It is not difficult to see that p, and -zzr are respectively functions 

 of x + y and x—y } for we have 



P + Q oc + y P-Q x-y 



2(1+0,) -(a + yf + fl* 2(1-^) («-y)« + *' 

 and in fact we find 



^ = v/T+26 log (# + y + \/(ff + y) 9 + « 2 ), 

 OT =v / T ::: 261og(^-y + v / (^-2/) 2 + « 2 ); 

 which gives the solution in the form 



(Q-(#+y +v /(»-fy) , +^ V ^^-y+v / (^^)M : « 2 ) Virir ) 



If in this differential equation we make x = x'\/ — 1, we have 

 the equation of the trajectory of a system of confocal ellipses 

 discussed by Mr. Boole at p. 246. The reader may compare 

 the modes of solution, and the forms of the result. 



16. Equations of the second degree which admit of being 

 placed in the form y = xfp-\-cf>p need not, of course, be inte- 

 grated by the formulae which I have given, as they are otherwise 

 soluble. The existing mode of solution, however, enables us in 

 these cases to determine the forms of p, and m ; and the use of 

 this process exhibits the complete primitive in some cases in a 

 more convenient form than the ordinary solution. Consider 

 the example 



p*-py-x = (Boole, p. 127), 



in which fp= , and cj>p=p. It is easy to find that 



Fp = (l + i)", and ®p= log (p- Vf-Y) ; 



and^? has the two values - (y± */y 2 + 4:x). With these we have 

 to form 



but as the functions of p Y and jo 2 are transcendental, we cannot 

 express finitely their symmetrical compounds. 



Making use of this solution so far as it goes, we find (to 

 adopt the method of this paper) 



dx ' dy y 



din fc d-& _ 1 +x //l +xY 2 



dx ' dy " y "V \ y ) 



2B 2 



