OF ARTS AND SCIENCES : JANUARY 14, 1873. 497 



The form of this equation shows that the value of the expression 

 b B is independent of the value of b ; it is, therefore, a numerical con- 

 stant, and may be denoted by e. 



i.e. b B = e ; 



B\og e b = 1, 



l 



whence B = 



loge b' 



Introducing this value of B in equation [4], we obtain 



D (log 6 xj*= , D l . 



K cb ' log e b. X 



Dx * 



If b = e, we have D (log e x) = — . [e?] 



a 



e is known as the Napierian base. The computation of its approxi- 

 mate numerical value is deferred until after the introduction of Mac- 

 laurin's Theorem. 



The logarithmic differentials of the Power, Product and Quotient 

 may be deduced by means of the above result, in the usual way, since 

 •the demonstration is dependent on the four elementary propositions 

 only. 



The Differential of the Exponential Function. 



Let z = x -\- h, then D z = D x, [1] 



also a z = a I+h = a h • a x [3] and D (a z ) = a h D (a x ). [2] 



Dividing equation [2] by the product of [1] and [3], we obtain the 



desired form 



1 D(a z ) 1 D(a x ) 



a z D z a x Dx 



Whence D {a x ) — c a x D x. [4] 



To determine c, we differentiate the identity 



x = log a a x , 



_ D (a x ) ca x D x 



tbus D x — \^r~ n — ^ = \zr~r. — ^ » 



loge a • a l°ge o. • a x 



C = loge a ' 



vol. vin. 63 



