of a New Class of Infinite Series. 263 
If we consider any series in which the first difference is 
equal to the digit occurring in the unit’s place of the corre- 
sponding term, as for example; the series 
6 6 
12 2 
14 4 
18 8 
26 6 
32 2 
a slight examination will satisfy us, that the value of the digit 
occur ring in the unit’s figure of w,, depends entirely on the 
value of w_, at the commencement op the series, and also that 
whenever the same digit again occurs, there will, at that point, 
commence a repetition of the same figures which have pre- 
ceded; consequently, the first difference at those two points 
will be equal. 
In the first example which I have adduced of a series of 
this kind, it will be found, that this reappearance of the ter- 
minal figure, happens at the 5th, at the 9th, at the 13th terms, 
&c. or that 
Au, = Ad; =A, =(Ad; =<. 
This gives for the equation of the series, 
Au,= Atl. 4 
or by integrating 
UW, = Us 4 + d, 
but when z = 1, wu, = w, therefore b = 0, and 
u, 41 — u, =e 
whose integral is 
u, = a(—/—1) + hag Hers icles 
d(—/—l) Sees Sie 
The four constants set determined, by comparing this 
value of w, with the first four terms of the series, we shall find 
Bee ees aa oe ee Y¥—-l,d=3+/7-1, 
and the value of «, becomes 
u,= 5(z—1) 4.63 —V—1) (VW -1¥ +(5 + V1) (—V — 1) 
which expresses any term of the series 
2, 4, 8, 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 dep 
solely 
