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XXI. Differential Equations of the First Order. Extension of 

 Integrable Forms. By Charles James Hargreave, LL.D. 

 Dub., F.R.S.* 



T ETm and v be functions of x, y, and j- (or p). I propose 



to designate u and v by the term "correlated functions" when 

 they are so connected that v [ -^u' (the accent denoting complete 

 differentiation with regard to the independent variable x) is a 

 function of x, y, and p only, and not of p'. 



It is well known that when u and v are correlated functions, 

 the differential equation 



v—fu, or <j)(u, v)=0 (1) 



is soluble by differentiation. This process gives 



™=fu, (2) 



where wis a function of x } y, and p, and is equivalent to v'-t-u'. 

 The division of v 1 by u 1 has the effect of expelling a differential 

 expression of the second order. If this expression be equated to 

 zero, it is a differential equation of the second order, the solution 

 of which gives the complete primitive of (1); but with this ex- 

 pression we are not here further concerned than simply to observe 

 that it disappears in the formation of (2). 



It is an elementary proposition that (2) is the singular solu- 

 tion of (1) when we substitute in it for p its value in terms of 

 x and y obtained by solving (1) with regard to p. It is equally 

 obvious (though I have not seen it noticed) that if we regard 

 (2) as a differential equation proposed for solution, its complete 

 primitive is (1), substituting in it for p its value in terms of x 

 and y derived from the algebraic solution of (2) with regard 

 top. The function/ as derived from/' necessarily introduces 

 the arbitrary constant which is essential in order that (1) may 

 be the complete primitive of (2) . All this may be thus shortly 

 expressed: — the eliminant of (1) and (2) with regard tojois the 

 singular solution of (1) and the complete primitive of (2). 



It is an immediate consequence of the relation which subsists 

 between u and v, that the equation 



<f>{u, v,w) = 



(where u and v are any two correlated functions of x, y } andp, 

 and w is v'-r-u') is integrable whenever 



is integrable. 



* Communicated by the Author. 



