T A Y 



129 



T A Y 



The problem is as follows : given 



y=V fjf+xfa) ....(A) 

 required the expansion of 4y< when possible, in powers of 

 x. Since tyy is, by the preceding equation, a function of 

 x and z, if z be constant, and we differentiate with respect 

 to x, and then make :r=0, or y=7z, we may use Stirling's 

 theorem. But this differentiation would be laborious and 

 indirect ; it was made more direct (by Laplace) in the fol- 

 lowing manner: A constant may have any value given 

 to it, or may be made to vanish, either before or after 

 differentiation with respect to a variable : if then we can 

 express differentiations with respect to x in terms of dif- 

 ferentiations with respect to z only (in which x is constant), 

 it will be in our power to make x vanish before the dif- 

 ferentiations, which will reduce the indirect or implicit to 

 direct differentiation. This substitution of z-difterentia- 

 tions in place of those of x is done as follows: Differen- 

 tiate (A) both with respect to x and z separately, and we 

 have 



l -j- x = F' (z+x<j>y) {<f>y+x<j>'y 



whence 



Let u be a function of y only, that is, not of x or z ex- 

 cept as these variables are contained in y : then 



'/" //// itu ih/ it// (In 



j T =$'/-, f~ or -= = *v -7- 

 dy dx fj ily dz dx ry <lz 



From this equation only it may be shown (by INDUC- 

 TION) that 



*~ / _ du 



du 



as follows. Assume the preceding to be true for one value 



of n, and, since fq&y) X du : dy is a function of y only, let 

 it be dv : dy, v being another function of y. 



d u d /dp dy\ d'v 



a \dv 



do 



dy 



dv dy 

 '-Jr.> 



dz 



whence the theorem remains true after writing re-f 1 for n. 

 But it is true when n = l ; therefore it is true for all values 

 o(n. If then we make ,c=0, or y=Fz, which may be 

 done before the differentiations on the second side of the 

 equation, we have (u being \fy) 



Apply this to Stirling's Theorem, and we have Laplace's 

 Tfn'nrem, namely, 



y=F (z+X(fnf) gives ^y= 



"-W Hr+.&c. 



the general terra, 



<** J ["] 



Lagrange's theorem, from which Laplace generalized, 

 is the case in which Fa; =3; ; namely, 



y=z+xrfjy gives ^y= 



' &c> 



W 3 ) ^-IrVi 



rf""' / 



the general term rri| 

 dz I 



y=z+<f,z.x+- (lz - -3 H a^T gj +. &c- 



Lngrange's theorem leads to liurjiitunis Theorem (\>n-- 



MBted to tlic institute in 1796). The second is in fact ihe 



ami- as the fir.it, though very different in form, and arrived 



at independently. It is required, when possible, to expand 



P. C., No. 1501. 



4x in powers of $x. This might be done indirectly, by ex- 

 panding <W l x in powers of r, and substituting <f>x for a? 

 in the result. The form in which Burmanu obtained 

 Lagrange's theorem avoids the indirect process. Let <ia: 

 vanish when x=a, and let <j>x= (x ): xa", or x=a+<]&; . xa ;. 

 We can now employ Lagrange's theorem to expand $x in. 

 powers of <j>x, and we have 



Now the general term of this has for its co-efficient the 

 value of 



when x=a : consequently tyx, expanded in powers of <f>r, 

 is found by making x = a in the co-efficients of the powers 

 of <f>x in the following series : 



d ffx-a\' N\i r0.r-) 8 



^C-^-; f*);J -2- + &c - 



When in a function of any number of variables a-,, x,, Sec., 

 the variables are severally to receive increments h,, h,, 

 &c., the law of the development is best seen by the cal- 

 culus of operations. [OPERATION.] To change x into 



x+h is to perform the operation e , D being the sym- 

 bol of differentiation with respect to x : the condensed 

 form of the development now before us is 



B ..... 



where D,, D 2 , &c. refer to x lt x v &c. The general term 

 of the development is 



(#,, x } , &c.) 



which must itself be developed. It is not worth while to 

 pursue this case further ; we shall only observe that when 

 it is desired to stop, the remnant may be obtained by 

 writing in the last term #, + (?/, for a?,, x^ + Qh^ for a;., 

 &c., where 0, the same in all, is either or 1, or between 

 them. 

 The value of x which makes cf>x=Q is represented by 



V 



&c. : where a is any assumed value (the nearer the root 

 the better) and 4>, <', &c. represent </>a, <p'a, &c. This 

 series is obtained by common reversion from <f}(a+/i)=0. 

 For the forms which Paoli gave to this series, and also to 

 Burmann's, see Lacroix, vol. i., pp. 306-308. The pre- 

 ceding series has been used, as far as three terms, in the 

 article APPROXIMATION. 



All that precedes is found in elementary treatises, with 

 the exception of a few terms of the last series : we now 

 come to matter which has been hitherto only the property 

 of the well-read mathematician, but which well deserves 

 to be made as common as Taylor's Theorem. We refer to 

 ARBOGAST'S method of derivations. Few, even among 

 mathematicians, are aware of the power of this process, 

 which may perhaps arise from their taking Lacroix's ac- 

 count of it, instead of consulting the work of Arbogast 

 himself: the former has only exhibited it to show that it 

 may be reduced to processes of the differential calculus ; 

 and even the latter has so loaded his method with heavy 

 applications, that he has concealed much of its beauty and 

 simplicity. 



The foundation of Arbogast's methods is a contrivance 



for expediting the expansion of <f>(a + bx+c.r i + ', into 



a series of the form A-f B.i- + C.t*+ The process by 



\\hich B is formed from A, C from B, &c. is uniform, and 

 is called derivation ; and A being (f>a, B may be called 

 D0, C may be called DD<a. or D s <a, and so on. Hence 

 h ought to be called Da, C ought to be D'-', and so on. 

 This notation is not precisely that of Arbosrast, but will 

 do lor our purpose. For more detail,* see the Differential 

 Calculus (Libra/ y of Um-jtd Knowledge), pp. 328-3'H. 



There It a (.Tent ili-nl nil tlia silliji-ct in tin- ' MulhiTnaticnl Trmt ' (pr,|. 



' at Kdi.ibarjri in li .* Mr. \W l 



. 

 humuu*) of the Rev. John West, I'ui 



VOL. XXIV. S 



