234 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



X 



li-^i 



n— 1 *=» 



The order of magnitude of such a function will, therefore, evidently be that 

 of the lowest term it contains; as it would be that of the highest term, if the 

 function, in place of vanishing, became infinite for x = a. And it will equally 

 follow from the definition, that very different functions may have the same 



order of magnitude: as, for example, the function x, sin x, , which, for 



1 + X 



X = 0, are all of the same order, and may be reduced one to another by omit- 

 ting parts that are of the higher orders. 

 An equation, 



P X + P X + &c. = Q Z + Q Z + &c. (7) 



11 22 11 22 



between two series of such terms can be shown to be impossible, unless the 



lowest terms are of the same magnitude. For, if not, we might suppose either 



term, as that on the left hand, to be the greater, and as by division we should 



have 



X z z 



P + p. A + &c. = Q. JL + Q J_ 4- &c. 



1 2 X 1 X 2 X 



1 1 1 



and as this for x = would reduce to 



P = 



1 



the lower terms would disappear until the required equality of magnitude is 

 obtained. 



It will be readily seen that we cannot regard the terms X, X, Z, Z, &c., of 



12 12 



the se9'ies as if they individually consisted of a single term ; since, even when 

 that is the case, we might always substitute for X or Z equivalent expressions 



n n 



that contained more than one term; and thus, to speak clearly, we must dis- 

 tinguish the total function, or term of the series, from the secondary functions 

 or terms, which, together with their corresponding primary function, constitute 

 X or Z ; and we must bear in mind that however this primary function is 



changed by expanding X or Z, or by substituting equivalencies for them, it 



n n 



always remains of the same order; a condition which is not requisite in the 

 secondary terms, which are, moreover, always of higher orders than their 

 primaries, which last express the order of the whole function, or term of the 

 series. 



