to the Computation of Mathematical Tables. 209 



which is one of a class of equations never hitherto integrated. 

 I succeeded in transforming this equation into a more analy- 

 tical form : but still it presented great difficulties ; I therefore 

 undertook the investigation in a different manner, and suc- 

 ceeded in discovering a formula which represented its ??th 

 term. It is the following : Table. 



u^ = (a) + 206 (106 + 2a — 1) 



where [a) represents the number opposite a in the annexed 

 subsidiary table, and a is the figure in the unit's place of z, and 

 b is that number which arises from cutting off the last figure 

 from z. Example : let the 1 7th term be required, then s= 17, 

 and a =i 1, b = \; the number opposite 7 in the table is 



{J)= . . . . 76 

 106 + 2a — 1 = 10 + 14 — 1 = 23 



206 = 20 206(106 + 2a — 1) = 460 



536 = u 

 17 

 I have formed other series of the same class, and have suc- 

 ceeded in expressing any term independent of all the rest by 

 two distinct processes. Thus I have incidentally been able to 

 integrate the equations I have mentioned : I will just state 

 one otlier of a simple form ; it is the ecjuation 

 Am. units fig. u 



whose integral is m. = 20b + 2<» 



where a is that one of the numbers 1, 2, 3, 4, which taken from 

 z leaves the remainder divisible by 4, and b is the (juotient of 

 that division. 



D <1 One 



