PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 277 



Now ^^^—r is the solution of the equation i/j^-a V^-l x -^ = X + ; which 



(Class II, Ex. 8, Cor. 1) is 



i_ A _ _L 



and a similar equation results for /3. Hence the solution of the given equation 



is known. 



18. It must be remarked of this solution, that it is not in aU cases complete 



without the introduction of the complementary (or arbitrary) function. This 



arises from the circumstance that when y contains positive integral powers of x, 



if- II d- rf* 



— ^ = 0, whereas x — . x — r " is not equal to 0. 



dxi dx^ dx^ ^ 



Hence a^ -j^ + ixi/ can be replaced by the latter function only by the con- 



d- x" 

 vention that — — r is not to be written when n is a positive integer. 

 dx^ 



On account of this convention, the solution of the equation ^ ^^ y=x must 



contain, besides the expression given for it above, a series of positive integral 

 powers of a;; and hence y, the solution of Equation (2), is incomplete without the 

 addition of such a function. It is probable, however, that the determination of a 

 relation between the arbitrary constants may give a solution possessing all the 

 generality which the science is capable of. We have already given an example 

 of the mode of avoiding arbitrary functions by introducing such a relation in 

 Example 1. We shall offer another as a corollary. 



Cob. If X = ~, the solution is (Class 2, Ex. 8, Cor 2.) 



^=^° + «^(;^-^) -a-3^(i-S ^arbitrary function 

 =y„ +- — (a + |S} - — + arbitrary function 



=yo + ;^-(« + ^)^—+px + gx''-+&c. 



Now if we examine the equation which connects together^, q, &c., we shall 

 find that it is the same as that which determines pi in Class 2, Ex. 3, having only 

 2 a in place of a^. Hence it is contained in the solution of the given equation 

 when b and X are omitted. It is, therefore, itself only a supplementary term 

 in the solution of the given equation, and its 4)lace may be supplied, appa- 



VOI,. XVI. PART III. 4 A 



