T A V 



128 



We call the preceding D'Alembert's proof, but it is 

 rather D'Alembert's result, and even tluit in a different 

 form : his real process is as follows: To take a case, in- 

 tegrate < (j+A) four times with respect to A. beginning 

 at A0: the results are (x+AcX for abbrevia 



and from this sort of proc. ilt is 



.1- + A) = 



(he two sides presenting the most identical forms which 

 have yet occurred. The integral may easily he reduced to 

 the form already given (.Lacroix, \ol. iii.. -p. :i'.!7 . U'Alem- 

 In-rt finished with the preceding form : it was Lagrangc 

 who first gave the limits which we have appended 

 above. 



Lagrangi's Proof. By this we do not mean the falla- 

 cious proof referred to in FUNCTIONS, THKOKY OF, but 

 that by which Ijigrange established the limits of the vnluc 

 of the remnant, which, on the ordinary definition of a 

 differential coefficient, is a proof, and a' very satisfactory 

 one, of the whole theorem. It rests upon the proposition 

 that if a function of .r have always one sign from x = <i to 

 / + A, the integral of that function taken between 

 UIOM- limits will have the same sign. 



If then we wish to establish Taylor's theorem as far as. >ay, 

 the term involving h'\ and to give the limits of the remain- 

 der, let P and ;> be the greatest and least values of <j> (<i+.) 

 from z = to s = h. Between those limits then <f> (a+r) 



same conditions, and we learn, step by step, that 



-/ -<", .s-'u - - P 



are severally negative. Hut . ;; is positive from .r 

 = a to :r = a + li : con-e(jUfi;t!y, proceeding in the same 

 manner, we find that, r being not greater than /i, 



is positive. If then we make s - li, we find that 

 In-- between 



<, + ,//.;,+ ....+ P. ~- and 



A* 



and the rest is as in the last proof. 



There is a proof given h) M. Cnuchy which resemble- 

 the preceding in its principle, though of very dit'eivul de- 

 tails, which nrny be seen in tin- Lib. ('. A., Ililferential 

 Calculus, pp. OS, &c., 7'>7. Hut this proof, though very 

 well iii a treatise on the subject, on account of the col- 

 lateral uses of the preliminary theorems which it requires, 

 is m,t so well suited to an isolated article on Taylor's 

 theorem. 



///// ire'i I't'-f. Let 4>r = <Jxi + P(.r a) ; differentiate 

 ively with respect ton, and we have 



= 4>'a + P' (x - ) - P 



= <{,"<, + P" (.,._,,)_ 21" 



= <i>"'<i + P'" (x - a) - 3P", &c., 



substitute for P, P', &c. their values : that is, substitute 

 from each equation to the preceding, and we have, milking 

 a + A, Taylor's theorem with Hie following result lor 



the remnant following the term which has A in it 



J" />.r - <JM\ 

 (lu*\ x -a ) 



making .r = H + A aft IT differentiation. 



une trouble to show the limits of this c\ 

 -ion. Ibi- \\liic-li we may refer to Ampere, IV-cU tie Cali-iil 

 Ditterculiel.' N:i-., .lourn. Kc. I'olytcchn.. t-ali. xiii., i>. IJi. 

 This trart nf Ampere i-. one of tin- purest deductions extant 

 of the Differential Calculus from the theory of limits. 

 In looking through all the proofs which give limr 

 it will he seen that neith. 



nor any dili'civntial coefficient employed can i 

 to become infinite between .r = n and .r=<i+/i. \ 

 such a circumstance dors occur, the theoivm ivlati. 

 the limits may ctase to be true. For instant 

 = (.r m)"" 1 , and stop the series after the first term, which 

 gives 



_ 1 _ 1 _ .1 A 



a + A m a in (a+0A in/' 



if a+A and a be both greater or both less than m. a value 

 of lying between (I and 1 will IK' found to 

 equation, as it should do from the theorem. Hut if '.I- = HI 

 between r=ti and .r=a+A, none but an imaginary 

 of will satisfy tliis equation. 



Stirling's theorem, a-, it should be called, Maclauri:: 

 it is called, is found simply by making =U in the d. \, 

 lopment of <f>(a+x). It gives 



B being either or 1, or between them. Here ^/"'(l means 

 that tf>x is to be differentiated ;/ times, and .r made =1) 

 after all the (lifi-reiitiation.i. This is the most useful form 

 of Taylor's theorem, with which it may be consider 

 identical in one point of view, and of which it is a parti- 

 cular case in another: for -i ../+./ , absolutely developed 

 by Stirling's theorem is simply ^ (+.) developed limn 

 ifn by Taylor's theorem. 



.loiiu Hernoulli's theorem, as given in the Leipsic acts 

 for 1090, is as follows: 



Here is an instant-e very much resemblinr the eonn. 

 of the UINOMIAI. THKOUKM ']i. 41'J with \\'alli<'s pn 

 investigations. If \Vallis had looked at his own result IM 

 a new point of view, lie niijjht not have left the- binomial 

 theorem for Newton: if John liemoulli had done (In- 

 same, he miijht ha\e s;iven the law of development of 

 0(.r+/i . Tin' ]',i ceding i-- a case of Taylor's them tin, :i . 

 follows: by that theorem 





l"J L"T'J 



and x Bx is the .-a me in meaning as 0.r, an undetermined 

 fractional part of .r. Let ifr.r = f'^jcil.r, then ., 

 substitution and transposition 



x dx = ^r. x- V* 



J,- x -, 

 [>'] 



This theorem is not of much use as a method of deve- 

 lopment, so that we need say no more of it in the present 

 article. 



Some views of Lambert on the reduction of the roots of 

 equations (Actn lli'lri-lii-ii. 17">s into series were jrene- 

 rah/cd by I.a^'raiiL'c (.W.m. ,-lrni/. Sri., 17(W, into a cele- 

 brated theiiiein of develojmieut bearint; his name; and 

 this again was generalized m form by Laplace (Mec. Cfl. . 



