This, if the reasoning be carried ton terms of Taylor's series, is Taylor's theorem with Lagmnge'a theorem on the remainder of the series appended. If no differential coefficient of ,px up to should become infinite from x= a to x=a +h., then 4'(a+h)=0a+0'a.h+ a 2.3...n-1 + 4'(") +B11) 2 . 3 ... n' If the remainder term diminish without limit as n increases without limit, Taylor's series gives a true development.
Some views of Lambert on the reduction of the roots of equations (Acta Helvetica, 1753) into series were generalised by Lagrange (Mem. Acad. Sci., 1763) into a celebrated theorem of development bearing his name; and this again was generalised in form by Laplace (Mee. Cel.). The problem is as follows: given y = F + . . . . (A) required the expansion of 4y, when possible, in powers of x. Since tky is, by the precediog 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 = FZ, we may use Stirling's theorem. But this differentiation would be laborious and indirect ; it was made more direct (by Laplace) in the following 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 differentiations with respect to z only (in which x is constant), it will be in our power to make x vanish before the differentiations, which will reduce the indirect or implicit to direct differentiation. This substitution of a-differentiations in place of those of xis done as follows :—Differentiate (A) both with respect to x and z separately, and we have dy dx dy--.-e(z+xfpy)111+xiiy, dx-- }- whence dy , dy dy dy (Z 1 + Y = Ti;Let u be a function of y only, that is, not of x or z except as those variables are contained in y : then du dy du dy du du Ti; d.c = (py d r o z ex = " crz From this equation only it may be shown (by Ixnvortos) that (lu\ = clz as follows. Assume the preceding to be tine for one value of n, since x du : dy is a function of y only, let it be dv dy, v being another function of y.
utdv dy v dx" = dz" (dy drJ = dz" d.+Iu {dv dv dv dy = or csy or cpy or 011 (0Y)* dz du dy, or (41/)"4-1 dZ whence the theorem remains true after writing + 1 for n. But it /6 true when n=1 ; therefore it is true for all values of n. If then we make x= 0, or y =rz, which may be done before the differentiations on the second side of the equation, we have (a being r d. 4,y (x = 0) = r (fprz)• dz Apply this to l?faclaurin's Theorem, and we have Laplace's Theorem, namely, y = F + xcpy) gives = dsPFz) d dOz) z + ((OF Zr + , &c.
dtprz the general term, (OF:). — dz [n] Lagrange's theorem, from which Laplace generalised, is the case in which FX= X ; namely, y + xoy gives Cy = cpz + 'z el + — clz (441 i" &e.
r x• the general term = 2 4- X + Lagrange'a Theorem leads to llurveann's Theorem, (presented to the institute in 1796). The second is in fact the same as the first, though very different in form, and arrived at independently. It is required, when possible, to expand in powers of ox. This might be done indirectly, by expanding in powers of x, and substituting st)x for x in the result. The form in which Burmann obtained theorem avoids the indirect process. Let ox vanish when x=a, and let cpx = (x—a) : xx, or x=a + tpx xx. We can now employ La grange's theorem to expand 4x in powers of tpx, and we have d \ , syx = xa+ra c•-2.
Now the general term of this has for its coefficient the value of - a ) {(--;p7 ) vx} when x= a consequently 4,x, expanded in powers of ibx, is found by making x= a in the coefficients of the powers of ox in the following series :— r x a r d - a \2 \ (4.42 &c.
When, in as function of any number of variables x„ x„ &c., the varia bles are severally to receive increments h„ h„ the law of the development is best seen by the calculus of operations. [OPERATION.] To change x into x + It is to perform the operation s", n being the symbol of differentiation with respect to the condensed form of the development now before us is e h,e, + + • • • • xe, where D„ &c. refer to x„ &c. The general term of the develop ment is [71 O(roXv &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 x, + Oh, for x„ x, + Oh, for &e., where 0, the same in all, is either 0 or 1 or between them.
The value of x which makes ¢x=0 is represented by — 0' 2 2 . 3 . 4 { + + _4'"4' 2. 3 .4 . where a is any assumed value (the nearer the root the better) and &c. represent cba, ora, &e. This series is obtained by common reversion from cs (a +11) =0, For the forms which Paoli gave to this series, and also to see Lacroix, vol. i., pp. 306-308. The preceding series has been used, an far as three terms, in the article