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JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY Communicated by E. H. Hall, April 22, 1921 The application of the Wilson-Sommerfeld quantum conditions to a conditionally periodic system with orthogonal coördinates involves the evaluation of an integral of the type The integral is to be extended over a complete cycle of values of q, which oscillates between two roots of f(q). The sign of the radical is to be the same as that of dq, so that if a and b denote the roots of ƒ(q), the integral can be written If f(q) is a polynomial of the second degree in either q or 1/q the integral can be cleanly evaluated. Otherwise, approximations are generally necessary. If f(q) can be expressed in the form where (q) is quadratic in q or 1/q, a is constant, and a (q) is small, a natural method of procedure is to try to develop J into a power series in Thus α. J(0) and J'(0) are easily evaluated, but unfortunately the higher derivatives of J with respect to a cannot be calculated by the usual methods because the higher derivatives of Vfq) with respect to a become infinite at qa and q = b. Hence this method is useful only when the higher order terms are negligible.1 Another method of attack employed by F. Tank2 and accepted as valid by others3 turns out on close examination to be faulty. Tank de velops f(q) into a power series about its maximum point, q = d. Let = q-d, and let H denote the maximum value of f(q). Then f(q) can be thrown into the form ‡ Tank in effect integrates this series term by term between the limits -(H/a) and +(H/a) and identifies twice the sum of the series so obtained with J. This procedure is wrong, since the correct limits of integration for J are ad and = bd. Moreover, the expansion is not usually convergent throughout the interval of integration, so that it is not possible to correct Tank's work by altering the limits. Another scheme of series development may be suggested, which avoids the above difficulty. Let the quantity u be defined by the equation H = √ H− f(q) = § √ a + BE + YE2 + 8 §3 +.... γ (5) same as that of §. The integral J can now be The sign of u is to be the thrown into the form Let us assume that de/du can be developed into a power series in u. Thus (7) (8) To evaluate K, we introduce the variable of integration 0 defined by the relation u H' sin 0. = It is easy to show from (9) by the application of well-known formulae that K, vanishes for odd values of 7; that K。 is TH; and that for even values of 7 greater than zero K, is given by the equation We proceed to the evaluation of the coefficients ao, a1, a2,.... As d¿/du may be regarded as a function of either & or u, let Then de/du = 0 = 0. The Since u and έ vanish together, the derivatives of at the point u can be calculated from the derivatives of x at the point & following method of procedure is perhaps the simplest. Let In some cases these successive derivatives of are simple and easily calculated functions of . In others the successive derivatives rapidly become complicated. If f(q) is given as a power series in the process of differentiation can be performed conveniently as follows: |