of a New Class of Infinite Series. 223 
The four constants being determined, by comparing this value 
of vu. with the first four terms of the series, we shall find 
a=0, b=-5,c=$-V-1,d=5+V-1, 
and the value of vu, becomes 
u, = 5(z—1) + §-— V—1)(V-1° + § + V—-1)(- V-1);, 
which expresses any term of the series 
9, 4, 85 16, 22, 24, 28, 36; 42, 44, 48. 
It is necessary, for the success of this method, that we should 
have continued the given series until we arrive at some term 
whose unit’s figure is the same as that of some term which has 
preceded it: now if we consider that this figure depends solely 
on that of the one which occupied the same place in the pre- 
ceding term, it will appear that the same digit must re-appear in 
the course of ten terms at the utmost, since there are only ten 
digits, and that it may re-occur sooner. The same reasoning is 
applicable to the case of series whose first difference is equal to 
any multiple of the digits found in the unit’s place of the corres- 
ponding term, or to those contained in the equation 
Au. = ax (unit’s figure of w.), 
as also to those in which this is encreased by a given quantity, as 
Au, =a (unit’s figure of uv.) + 0. 
If the second difference is equal to some multiple of the figure 
occurring in the unit’s place of the next term, as in the series - 
Qe DAD NONALO, 
already given, since the unit’s figure must always depend on the 
same figure in the first term of the series, and its first difference 
2 ty) 
10 
