of a New Class of Infinite Series. 219 
perceived, that the terms of the latter recur after intervals of 
four, and that all the changes in the first differences, are com- 
prised in the numbers 2, 4, 8, 6, which recur continually, and 
the series may be written thus: 
Series. Diff. 
2 2 
4 4 
8 8 
16 6 
5 22.= 90 -+- 2 2 
24 = 20+4 4 4 
28 = 20+ 8 8 
36 = 26 + 16 6 
42 = 40+ 2 2 
10 44 =40+4 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 
82 = 80 + 2 2 
If then z be of the form 4v + 7, 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, 
u, = 20v-+ 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., 
EE2 
