326 NEW SERIES for the 



indifcriminately ; therefore B =z Nap. log b, and confequently 



log x (to bafis b) = Nap.logx 

 6 v J Nap. log b' 



as is commonly known. 



58. Since, therefore, the logarithms of any propofed fyftem 

 may be deduced from thofe of Napier's fyftem, I ihall 

 throughout the reft of this Paper attend only to the formula 



Nap. log x — n (x — 1), n being indefinitely great. 



Let us then, agreeably to the mode of proceeding employ- 

 ed in the former part of this paper, afTume the identical equa- 

 tion 



X a + 1 _ X+i , 1 X—i 



X 2 — I ~ 2(X— l) "*" 2 X+I* 

 _£, ' i • 1 1 



In this expreffion let x 2 , x+, x*, x ,z , &c. be fubftituted fuc- 

 ceffively for X, and let the refults be multiplied by the corre- 



11 /* 



fpondinff terms of the feries, 1, -, -, &c. Thus there will be 



24 



formed a feries of equations, which, putting n for their num- 

 ber, and m for 2 n , may ftand as follows : 



x+ 1 



x — 1 



