the Theorems of Maclaiuin and Taylor. 99 



the brackets indicating what the expression between them 

 becomes when cr = 0. 



In writing down Maclaurin's theorem, authors differ con- 

 siderably as to the notation employed for the coefficients. I 

 have thought it would be well, for the sake of uniformity in 

 this respect, if the plan generally adopted for distinguishing 

 the coefficients of Bernoulli, and which, indeed, are only those 

 of Maclaurin in a particular case, were universally followed. 

 Maclaurin's theorem would thus be written 



F(« + .r) = Mo+MiX + M2^+M3|^+&c., . (1.) 

 and Taylor's, 



F(^- + «) = To + T,« + T,|VT3g+&c. . . (2.) 



Applying the first of these theorems to the function y = \og 

 {a -f-.r), in order to illustrate the foregoing principle, we have, 

 making c^?=0, 



&c. &c. 



.•.log(« + ^)=iog«+^-^ + ^-^ + &c. 



And, from the process by which the coefficients have here 

 been deduced, it is plain that Taylor's development of lo^ 

 [x-\-a) is got from this by simply changing the places offl and 

 x\ and that the one development is always convertible into 

 the other, by thus interchanging the two terms of the binomial 

 under F. There is therefore no necessity for considering (1.) 

 and (2.) as distinct theorems: nor is it correct to do so, since 

 they are both comprehended under the same form ; for which- 

 ever be the term according to whose powers the development 

 is to proceed, the coefficients are always to be derived from 

 the function after that term is made zero. Hence, calling either 

 term, indifferently, /; and denoting the differential coefficients 

 derived, as here explained, by D,, Dg, &c., the following 

 theorem comprehends both (1.) and (2.) : 



y=Do + Di< + D2| +D3^ + &c. 



Even when the function is not that of a binomial, but of a 

 simple monomial, as a', log*', &c., still it will be advisable, in 



H2 



