462 ON THE INTEGRATION OF CERTAIN DIFFERENTIAL EQUATIONS BY SERIES. [532 
and then 
a+2 a _ 2ar + 6a' J — 4 _ 2 (a + 1) (a 2 -F 2a — 2) 
a(a + 3) + (a+ 2) (a — 1) (a—l)a(a + 2)(a + 3) (a—l)a(a + 2)(a+3)’ 
so that the whole expression contains the factor a + 1. But observe that in the 
present case, if (as is allowable) we write D = 0, the next coefficient E (depending 
not on D only, but on D and G) will not vanish; so that the solution obtained on 
the assumption D = 0 will go on to infinity: and if instead of assuming D = 0, we 
assume D = an arbitrary quantity D', then E and the subsequent coefficients will 
contain terms depending on D'; and the complete form of the series belonging to 
a — — 1 will be 
y = A. -f- it; 2 H— it? 3 + &c.^ + I) ^¿c 2 + y x° 4* &c.^ , 
where the second member is in fact the series belonging to a = 2. It is hardly 
necessary to remark that the solution thus obtained can be expressed by means of 
exponentials, viz. that the solution is 
y=A 
1 9 
e 6x + D' , 
q° 
e ex .