THE ANALYST. VOL. III. MARCH, 1876. No. 2. NEW DEMONSTRATION AND FORMS OF LAGRANGE'S THEOREM. THE GENERAL THEOREM. BY LEVI. W. MEECH A. M. HARTFORD CONN. "In the case of Lagrange's Theorem, Lambert of Alsace (died 1777), in endeavoring to express the roots of Algebraic equations in series, found a law resembling that which we have just developed. He published his results in 1758, and Lagrange generalized them into the theorem, which bears his name, in 1772. Finally, in the Mecanique Celeste, Laplace made a still further extension." (DeMorgan's Calculus, p. 171). ! The following investigation from a new point of view, appearing entirely conclusive, is presented to facilitate the study of analysis. At the close, the reasoning unexpectedly leads to the general Theorem, under a simple expoential form, which includes this whole class of Theorems, especially those of Taylor, Lagrange and Laplace, as particular cases, according to the initial differentiations there described. (1) To find the value of x in terns of t and e from the primary equation, x = t + efx = t + h. Let us make the single letter h = efx, as already denoted; then taking the same function ƒ of each side, (2) = fx = f(t + h). The first member fr h÷e, as indicated above. And this equals the second member developed in powers of h by Taylor's Theorem: (3) (4) dft.h + + ... dt The auxiliary h can now be found by simple reversion of series; thus let, h=Ae + Be3 + Ce3 +De+.... Substituting this series in place of h in equation (3) and equating the coefficients of equal powers of e; also denoting ft by the single letter ƒ, (5) h = ef + + + + ... ďf 1 df1e 1 dfe 1.2 dt 1.2.3 dt 1.2.3.4 de The regular law of the series continues to any extent, as will be shown presently. Substituting in equation (1) we have a particular case of Lagrange's Theorem: Generally, taking any function of each side of equation (1), and develop Here, substituting for h its value from equation (5), and reducing, we have Lagrange's Theorem: It only remains to prove that the regular law of the first four terms of the series, continues uniform. Developing each member of the last equation by Maclaurin's Theorem, and equating coefficients of equal powers of e, The simple law of third Maclaurin derivation by increasing the exponent of derivation and of ft, each by unity, is obvious. And since it is applicable to any form of the function F, let a new function F' be assumed, such Differentiating these equals successively, we see that the law of the advance in Fr must coincide with the law of the rear terms already established in both Fx and Ft; and so on to infinity. Equation (6) is included, by making Ftt. Note 1. In certain examples, where eft contains t also, the demonstration plainly teaches us to regard such t as constant. That is, to differentiate t only where it has taken the place of x; or rather, to differentiate with respect to x, and change x to t afterward. Note 2. To derive one part from another, let A, denote the n' term of the formula (6) or (8) to commence with the second term; then, Example 1. In the regulur application of the formula to equations, Lagrange found that it always gave the least root. Let it then be applied to give the least root of the common quadratic. Here x = a + bx2; by 'formx = a + ba3 + 2b3a3 + 5b3a* + 14b*a3 +...... ula (6), The convergence of the series evidently depends on the smallness of ab. Example 2. It is required to express the third power of the least root of the quintic = a + bx3. By formula (8), Fx = x3, dfx dx = 3x3, efx = bx3. x3 a3 + 3ba7 +18b3a" +13663a15+.... Example 3. To find what number is equal to ten times its natural or hyperbolic logarithm. In the primary equation, t = 0, x =0+ e12. By (6), 0.2 (0.1n)"-1 x = 0 + 1 + + n! ... + = 1.118325. Example 4. If a denote the left hand digit or digits of a number and b those of the logarithm, it is required to find the remaining digits such log (b + .01x), or such that the right hand digits of both number and logarithm shall be alike. that a +x= Lagrange's Theorem for several variables. Let efx = h, e'f'y = h',... in the primary equations, (11) The value of h, of h', .... can each be written out separately by the preceding equation (5). It only remains to substitute these series in the required function (12) after its development in powers of h, h', .... by Taylor's Theorem for several variables: The result being independent of the order of operations, is better stated in (26). Laplace's Theorem. The primary equation may have the more general form, (13) x=f'(t + efx) = f'(t + h). Taking the function ƒ of each side and developing the powers of h by Taylor's Theorem, (14) fx = h = ƒƒ't + diff't.h + *ff't. 1.2 + .... e ff't Let this be compared with equations (3) and (5). Again, taking any function of each side of equation (13) preparatory to development by Taylor's Theorem: Let this be compared with equation (7). Whence it is obvious that Laplace's Theorem will be found by simply changing in (8), Ft into Ff't, and ft into ff't. That is, In the case of several variables, the changes in equations (12) and (26) are similar and obvious, if the primary equations have each but one unknown quantity as in (11). The more general equations will be investigated presently. Convergence. To render Lagrange's series more convergent, let a denote any approximate value of x-t or efx; as a beft, where b=1+d(eft)÷dt, nearly. Then if t′ =t+a, the primary equation becomes Again, the arbitrary nature of Fr or Ft can generally be made to give a more convergent series. For illustration, let Fx in equation (8) have the form below, where m denotes any assumed positive integer; and dft = Tdt, em+2 dm+1 1.2..(m+2) ̊ dtm+] ·(ft.mT)—... Observe whenever T=ft, one term vanishes by differentiation. Again, Fr may take the forms below, if integrable: (21) cot u=cott-e÷sin te3 sint + e' sin 2t Or multiplying by sin u sin t and reducing, (22) (23) (24) e'e-esint- Jesin't cost +e(sin't—sin)+... Whence The limiting values of this series are remarkably simple. For, taking the sine of each side of (20) after transposing t; multiplying and dividing by sin u; and comparing with (22), we find sin (e sin u) sin ue'. Making sin u = 1, we have sin ee', the least value of e'. Again, making sin u very small or 0, we find e = e', the greatest value of e'. Another solution. In (20), and the preceding expression, making sin u = 0, 0.1, 0.2, 0.3, ... the value of e' can also be estimated or interpolated from the following OUTLINE TABLE. t = 30°-0.5e 36°52'-0.6e 44°26'-0.7e 53°08'-0.8e 64°10'-0.9e 90°-e 10 sine sine 10 sine e2 sin te sine sine The value of e is always positive, also that of u, 180° ±u, 360° — u. Symbolic Terms. Expanding the logarithm below in series, developing each numerator by Taylor's Theorem till the power of the increment h is the same as in the denominator, we see by comparison with (8) that Lagrange's Theorem is represented by the terms independent of h in the development of the following expression: (25) Fx = Ft — h . log { 1 — f(t + h) }. dM(t + h). h d(t + h) This result is equivalent to the definite integral first found in 1805 by M. Perseval in imaginary exponentials. Again, if the symbol D=d÷dt, D' =d÷dt', D3=d3÷dt...formulas (8) and (12) may also be written as in (26). |