of a New Class of Infinite Series. 221 
Series. 1 Diff. 
1010 86h= 1 80) 41.6 
1096 92 = 80+ 12 
25 1188 100 = 80+ 20 
,» 1288 108 = 80 + 28 
1396 114 = 80 + 34 
1510 114 = 80 + 34 
1624 118 = 80 + 38 
30 1742 120 = 120 +. 0 
1862 122 = 120+ 2 
In this series it may be observed, that u. when z is less than 10, 
is equal to the sum of the first differences of all the preceding 
terms, and if « be greater than 10, it will be composed of four 
terms, viz., first the sum of the ten first terms of the first difference, 
multiplied by the number of tens contained in z; secondly, of 
the sum of the series 40 + 80 + 120 + to as many terms as there are 
tens in z, this must be multiplied by 10, as each term is ten times 
added; and thirdly, of the number 40 multiplied by the same 
number of the tens, and also multiplied by the digit in the 
unit’s place of 2; and fourthly, of the sum of so many terms of 
the series as is equal to the unit’s figure of z; this being expressed 
by (@) signifymg the number opposite a in the previous Table. 
These four parts, if 2 = 10h +a, are thus expressed, 
1* 1808, 
b.b-1 
ae 
3" 40ba, 
Pie 1 (7) 
These added together produce 
u,=20b(10b + 2a—1) + (a). 
This value of u., if diminished by 2, is equal to the sum of z—1 
term of the series which constitute the first difference. 
oe AO 
