98 BISHOP TERROT ON THE SUM OF THE DIGITS OF NUMBERS. 
If, therefore, the sum of the two numbers substituted for 2 in the expression 
a°+32+1, be 8, the differences in the two cases will be identical. 
Substituting successively 0, 1, 2, &c., we have the series 
ey 3! 2 3 4 5 6 7 8 9 10 11 12 
1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, &e. 
Differences/"1, 5, 40°98)" 7s; 0; bf Tey)? x0, \'o1, 5, &e., 
where we see, 1s¢, that the differences recur after 11 terms; 2d, that if 0 and 8, 
1 and 7, 2 and 6, &c., be substituted for a, the differences are identical. 
EDINBURGH, December 2. 1845. 
