142 H. S. CARSLAW. 



And Napier's final form is, of course, 



nl x = 10 10 log 10 a?, 

 Biirgi's Arlthmetische und Geometrische Progress Tabulen 

 also come under the same law. If the terms in the Arith- 

 metical Progression are taken as the logarithms of the 

 terms in the Geometrical Progression, and Bl oc stands for 

 what I may call Biirgi's logarithm of ce, we have 



lOge , 1Q8 j 



Bl * = 10 ^1 - = 10 log i+ , fa) 



lo g e(l + 1 |) 



for x = 10 s ( 1 +jtu) > s being any positive integer. 



Finally, treating Napier's series 

 0, 1, 2 



w,w(i-^),w(i-^f 



in the same way, and denoting this logarithm by Nl a% we 

 have 



Nl oc — 



log* 



10 V , /•* 



'oe 



1\ =l ° g l- 



10' 



lOffe ( 1 - i4r l0 



1 1ft< » 



10' 



for x — 10 7 ( l _ ^r) > s being any positive integer. 



