the Development of Ms Theory. 485 



equal differences, can be taken as the starting-point of the 

 theory ; and that, if the logarithm of unity is zero, the 

 logarithms of the product and the quotient of two numbers 

 are, respectively, the sum and difference of their separate 

 logarithms. 



The Differential Equation satisfied by the 

 Logarithm of x. 



§ 9. We have seen that the theory of the different systems 

 of logarithms described in the previous pages rests upon the 

 fundamental property : — 



If a : b = c :d, then 



X(a)— \(6)=X(c)— K(d), 



where \(x) stands for the logarithm of x. 



The function \(#), therefore, satisfies the equation 



\(a? + A)-X(^)=\(l+-)-X(l). 



h 



x 



Proceeding to the limit 7i-M3, of course keeping x fixed 

 we have 



\'(x) = -, where A = X'(1). 



Therefore X(a?) = A log m + B, 



and the system is made definite by adding two other 

 conditions. 



In Napier's Canon, writing p for the radius, we have 



nI#=A log# + B, 



with nlf> = 0, and nl'f>=— 1. 



Therefore , , / p\ 



n\x=p\ogJ^y 



In Briggs's modification of the system, we have 

 bl^ = Alog# + B, 



wifch bl^Oand bl^=10 10 . 



