98 BISHOP TERROT ON THE SUM OF THE DIGITS OF NUMBERS. 



If, therefore, the sum of the two numbers substituted for x in the expression 

 r^+sa + i, be 8, the dififerences in the two cases will be identical. 

 Substituting successively 0, 1, 2, &c., we have the series 



0123456789 10 11 12 



1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, &c. 

 Dififerences, 1, 5, 0, 8, 7, 8, 0, 5, 1, 10, 10, 1, 5, &c., 

 where we see, \st, that the difTerences recur after 11 terms ; 2d, that if and 8, 

 1 and 7, 2 and 6, &c., be substituted for oc, the dififerences are identical. 



Edinbuegh, December 2. 1845. 



