
PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. aT 
Now eae! is the solution of the equation y,-—aW/—1~2 a ib See ot 83 which 
(Class II, Ex. 8, Cor. 1) is 
1 1 1 
ds (act Age of fo ) 
(I+av= alias oe) Ma tn wae 
and a similar equation results for 6. Hence the solution of the given equation 
is known. 
18. It must be remarked of this solution, that it is not in all cases complete 
without the introduction of the complementary (or arbitrary) function. This 
arises from the circumstance that when y contains positive integral powers of «, 
Ps + 4 
ad) =(0, whereas 2 ey Asp uv is not equal to 0. 
da x dx dx 
Hence «? = + tay can be replaced by the latter function only by the con- 
Y a 
vention that = is not to be written 0 when n is a positive integer. 
5 : : é : 1 
On account of this convention, the solution of the equation Itap y=X must 
contain, besides the expression given for it above, a series of positive integral 
powers of 2; 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. 
Cor. If X= ie the solution is (Class 2, Ex. 8, Cor 2.) 

a “eB (eee ge) aa (4-5 
=y,+ = —(a + B) a3 + arbitrary function 
) +arbitrary function 
Sak oe + 8) qe Pete + he. 
Now if we examine the equation which connects together p, g, &c., we shall 
find that it is the same as that which determines y, in Class 2, Ex. 3, having only 
2a in place of a2. Hence it is contained in the solution of the given equation 
when 6 and X are omitted. It is, therefore, itself only a supplementary term 
in the sclution of the given equation, and its place may be supplied, appa- 
VOL. XVI. PART III. 4A 
