WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION? 25 



entry: a biordinal equation to a group of double entry, out of which an infinite number of 

 groups of single entry may be collected. Thus, b and c being in relation in <p(a!, y, b, c) = 0, 

 we may designate all the curves contained in ip{a;,y,fc,c) = as the group (fc,c). Generally 

 speaking, the curves of the group (fc,c) are different from those of (Fc, c). The unlimited 

 number of cases of (fc, c) is the key to the unlimited number of primordinal equations which 

 give rise to one and the same biordinal equation. It is then the characteristic of the biordinal 

 equation that it represents a group of double entry. When the constants are not in relation, 

 as in (h{x,y, b) . \//(a?, y, c) = 0, we have still groups of double entry, but the biordinal equation 

 ceases to exist : the distinction between one group and another consists in the distinct ways in 

 which individuals of the two groups ^ = and \|/ = are joined together. This defective 

 grouping — not defective in the variety of its cases, but defective in the variety of the elements 

 out of which cases are to be compounded — is within the compass of a primordinal equation, 

 into which therefore the biordinal equation degenerates. 



As an instance, let (P - b) {Q - c) + R = 0, P, Q, R, being each a function of ai and y : 

 and let P' represent P^ + P^. y, &c. 



When b = fc, the primordinal equation of the group (/c, c) is 



^ R' + -^(R" + 4P'Q:R) a R'--y/{R^= + 4.PQ'R)[ 



y+ r^T — =J{P + 



2P' •'\ "^ 2Q' /■ 



Let R = fxV, where F is a finite function, and fx a constant. When /x diminishes without 

 limit, and finally vanishes, each primordinal equation becomes either P' = or Q' = 0, for 

 otherwise we have only Q = fP, the algebraic result of eliminating c between (P — 6) (Q — c) = 0, 

 and (P —fc) Q' + {Q - c) P' = 0. And the biordinal equation is determined by differentiating 



, ^ r'+^(r'^ + 4P'q:r) 



6 = Q + y^^ -. 



Do this fully, clear the result of fractions, and write /xV for R : it will then appear that y" is 

 seen only in terms multiplied by positive powers of ^t ; and so that n = gives P'Q' = in place 

 of a biordinal equation. 



The correction which the common theory requires is as follows ; — An equation in which n 

 constants are in relation with a? and y, cannot have any differential equation clear of those 

 constants under the wth order ; and an equation of single and irreducible relation between 

 w,y,y ,...y^''^ must have a primitive containing n constants in relation to ir and y. But a 

 primitive equation in which n constants are contained in alternative relations, Mj in one relation, 

 w in a second, &c. does not require a differential equation of the nth order ; but has an equation 

 of alternative relations, one of the Wjth order, one of the n^th order, &c. 



From a primitive having n constants, in relation with x and y, no constants can be 

 eliminated in favour of y', y", &c without one new equation of differentiation for every constant 

 which is to disappear. But this is by no means true of constants in relation with x,y, and one 

 or more of the set y',y",..., to begin with. This point is made clear enough in the section of 

 my former paper to which these remarks form a supplement : but the whole may be illustrated 

 as follows. If (p(at, y,a) =0 give a = (^{x, y,), and therefore 4>j + <I>y . y' = for a differential 

 equation, in which a has disappeared and y is introduced, it is easy to give this differential 

 Vol. X. Part I. 4 



