ELISHA MITCHELL SCIENTIFIC SOCIETY. 1 7 



dy = I lim. — Idv = ^x^dx. 



From this equation we see why 3.T- (in the particular 



\v 

 example) or lim. — generally, is called a differential co-efl5- 



cient. 



On referring to the right member of eq. (4), w^e see that 

 regarded by itself^ it has no limit, since it is an essential 

 requisite that a variable can never reach its limit, whereas 

 by making A.r ^ o, the right member becomes at once yx?^ 

 but considered in connection zcit/i the left member^ we see 

 that although A.r must tend towards zero indefinitely, yet 

 it can never be supposed zero, for then the ratio ^y -^ ^x 

 has no meaning. With this restriction, then, the right 

 member can approach 3.??- as near as we please without ever 

 being able to reach it; hence 3.^- is the true "limit" of 

 the right member when b.x is regarded as an infinitesimal 

 whose limit is zero. 



It is evidently immaterial by what law, if any, c^x di- 

 minishes towards zero. We can, if we choose, suppose ^x 

 to diminish by taking the half of it, then the half of this 

 result, and so on, in which case A.r will tend indefinitely 

 towards zero, but can never attain it; or we can suppose 

 A.i' to diminish, in any arbitrary w^ay, indefinitely towards 

 zero without ever becoming zero. In any case the right 

 member as well as the left has a true limit according to the 

 strict definition. 



It is to be observed, too, that this limit is found on the 

 one supposition that a.c tends towards zero, for then aji', as 

 a consequence, tends towards zero indefinitely without ever 

 beine able to reach it. 



■C5 



We have emphasized this point, because some of the best known Eng. 

 lish writers, as Todhemter, Williamson and Edwards, following the lead of 



