L 355 j 



LVIII. Differential Equations of the First Order. 

 By Charles James Hargreave, LL.D. Dublin, F.R.S.* 



Introductory Remarks. 



1. r I^HE object of this memoir is to call attention to a defect 

 J- in the treatment of differential equations of the first 

 order, and of the second or any higher degree, and to supply 

 that defect, so far as my researches enable me. 



With regard to this class of equations, the only general pro- 

 cess of solution given in elementary works consists in resolving 

 the equation into as many equations of the first degree as it will 

 admit of, and then solving each of these equations of the first 

 degree by any method which may be practicable for each of them. 

 An attempt is then made to combine all these distinct solutions 

 into one, by a process which I conceive to be an illusory one. 

 The result of this method, therefore, is, not the solution of a 

 differential equation of the ?zth degree, but the separate solution 

 of n equations of the first degree, which are perfectly distinct, 

 and incapable of being connected by any process arising out of 

 this method. In practice, equations of the second and higher 

 degrees are not solved by this method — except a few forms pre- 

 pared for the purpose in such a manner that the radical opera- 

 tions are capable of being actually performed, so that we obtain 

 equations of the first degree free from radicals. Of this cha- 

 racter are (p being -^-), 



p*- a y = (Boole, Diff. Eq. p. 116), 



p 3 — (# 2 + xy + y^p 2 + {x B y + x 2 y 2 + x^)p —a?y 3 =0 



(Gregory, Ex. 328), 

 {a i -x 2 )p 3 + bx{a 2 -x 2 )p' 2 -{p + bx)=0 (Idem), 



which admit of division into two or three simple equations not 

 containing radicals, and each easily integrable. The results of 

 these separate integrations cannot, however, be connected to- 

 gether, except by the fallacious process of making the three 

 arbitrary constants all equal, which is a logical absurdity. Take, 

 for example, the second of the above instances, and resolve it 

 into 



p — <2? 2 =0, p—xy = } p— ?/ 2 = 0. 



The three separate integrals are, or rather may be, 



* Communicated by the Author. 

 2 A 2 



