l6 JOURNAL OF THE 



thehi by these rules is comparatively simple, however com- 

 plicated the functions. 



A3' f{x + h) —f{x) 

 As lim. — , lim. , are cumbersome sym- 



A.r Ji 



bols it is usual to put — for them. 

 d.v 



dy Av /(.r + //) -f{x) 



. • . — = lim. — = lim. . 



dx ^x h 



In this expression dy is read differential of y and dx dif- 

 ferential of a;, and both dy and dx are to be regarded as 

 indivisible symbols, so that d \^ not a factor but a symbol 

 of operation. The differentials dy and dx are regarded as 

 finite quantities, whose ratio, for any value of .t, is exactly 



AJ/ 



equal to lim. — . 

 c^x 



Thus even for the same value of this limit, dy and dx 

 can be supposed to both increase or both decrease at pleas- 

 ure, the only restriction being that their ratio shall always 

 equal the value of the limit for the particular value of x 

 considered. There is thus great flexibility in this concep- 

 tion of differentials. As a rule we shall consider the dif- 

 ferentials as having appreciable values; in other cases it is 

 convenient to treat them as infinitesimals ox finite quanti- 

 ties whose limits are zero^ but which consequently never 



dy 

 become zero themselves, as then the ratio — has no sig- 



dx 



nificance. In the same way aa' and aj are infinitesimals. 

 In the equation, derived from one above, 

 ^ dy Ay 



— = lim. — = 3-^^, 



dx A.U 



it is understood that we can clear the equation of fractions 

 and write. 



