274 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
and so on if other terms of the factors are wanting. Thus we can in all cases 
obtain the transformed equation in the required shape ; but in all the cases where 
coefficients thus vanish, there is the important qualification that is no longer 
necessarily of the /ith order ; so that H does not necessarily contain the proper 
number of arbitrary constants. The consequences of this consideration will be after- 
wards developed ; in the meantime we proceed to consider what modifications the 
series undergo where f^t has two or more equal roots. 
12. If there should be tvm roots of f^t=0 equal to jSi, one series will be deficient ; 
and it will be supplied as follows. The expression H, or (/o(D))“'( 2 "®G), contains in 
that case a term of the form ; and it is easily seen that 
<p (D ) (^£^0 = ^^(D) (£^'«) + ?)' (D) (£^‘0 . 
The wanting series is, therefore. 
where 
{ log a: ( 1 + A,x + + . .) + ( A',x + A'ax" +..)}, 
a;=^&c. 
If there should be three roots of fot—O equal tojSj, two series will be deficient, one 
of which will be supplied as last mentioned ; and the other by the introduction of the 
series 
{ (log x) X 1 + A,x + A^x^ + . .) + 2 log x( A'jX + A; x" + . .) + (A jX A;'x" - f . .) } , 
where A'(=-t^^ &c. ; for we have a term /r/V'® ; and it is easily seen that 
api 
^(D)(^¥'^)=^^?)(D)£^'® + 2^9'(D)£^'® + ?3"(D)£^'^ 
And generally since, where there are p+l equal roots |3,, we have terms 
Ci £^'® + ; 
and since 
the deficient series will be supplied by the following, taken with a different constant 
for every value of p from unity upwards : 
/rp¥‘|(logx)'’(14-A,x+A2X^4---)+/'(lo§‘ x)^~'(AiXd-A2X®+..) 
+/?^Y^(logx)p-"(A"xd-A'>"+. .)+..-{-/? log x(Ai^"'^x+A 2 ^“'¥'+..) 4 -(A^,^¥+A 2 ^V+. 
As a matter of convenience, when the equal roots are zero, a temporary nominal 
value should be given to them for the purpose of differentiating Ai, &c. 
13. Hitherto we have attended especially to the complementary solutions, or in 
other words, regarded G as zero. The operations, however, indicated in (3.) may be 
readily performed when G is a rational function of x; which we will suppose to be 
