229 Mr. BaspaGe on the General Term 
This inductive process for discovering the 7» terms of such 
series, might be applied to others of the same kind, but it does 
not admit of an application sufficiently general or direct, te 
render it desirable that it should be pursued further. 
If we consider any series in which the first difference is equal to 
the digit occurring in the unit’s place of the corresponding 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 
occurring in the unit’s figure of w., depends entirely on the value 
of u., at the commencement of the series, and also that when- 
ever the same digit again occurs, there will, at that point, com- 
mence a repetition of the same figures which have preceded ; 
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 re-appearance of the terminal 
figure, happens at the 5th, at the 9th, at the 13th terms, &c. 
or that . 
This gives for the equation of the series, 
Au, = Au.+,4, 
or by integrating 
u,=U,44+ 5, 
but when z=1, w, = u, therefore b = 0, and 
U.,1—U, = O, 
whose integral is 
u,=a(— V—-1)*4+b(- V—1)**!40(— V—1)2+*4d(— VY —-1)**3+53. 
