-of a New Class of Infinite Series. 261 
Series. Diff. 
98 = 20+ 8 8 
386 = 26 + 16 6 
42 = 40 4+ 2 2 
10 44 = 40 + 4 4 
48 = 40+ 8 8 
56 = 40 + 16 6 
62 =60+ 2 2 
64 = 60 + 4 4 
15 68 = 60+ 8 8 
76 = 60 + 16 6 
S2r—— SU ey 2 
If then z be of the form 4v+2, the value of u, will be 20v + one 
of four numbers 2, 4, 8, 16, according to the value of 7, and if 
i always represents one of the numbers 1, 2, 3, 4, the value of 
u_will be thus expressed, 
uw, = 200+ 2. 
As a second example, let us consider the series whose first 
term is 2; its first difference 0, and its second difference always 
equal the unit’s figure of the next term ; its equation will be 
A*u, = unit's figure of w,, 
and the few first terms are 
2 28 
2 48 
4 76 
10 110 
16 144 
182 
This series may be put under the form 
Series. 1 Diff. 
¢) 2 0 
2 2 
4. 6 
10 s 
ae ue Table of (a). 
5 28 20 if a=0(d) =2 
48 28 1 2 
76 34 2 4: 
110 34 3 10 
144 38 4 16 
10 182 40 = 40+ 0O 5 28 
222 42=40+4+ 2 6 48 
264 46 = 40+ 6 7 76 
310 46 = 40+ 6 8 110 
356 52 = 40 + 12 9 144 
