Adding this to the equation L(1 − a) = fd; log (1 − a), To determine the constant, make a=0, when as log (1) = 0 and L(1) = 0, we have L2 (x) + L2 (1 − a) = log (x) . log (1 − a) 2 6 2 This property of the transcendent L2 is only true so long as a is less than unity, as when any greater value is assigned to it log (1 − a) becomes impossible. Hence L(1 − a2) = 2 L. (1 + x) + 2 L (1 - x). L¿(1 − (9) If we take the upper sign, and in therefore L2 (1+) + L ̧(1 + x) = § (log x)2 + L2 Analogous properties may by the same method be demonstrated of L2 (1 ± x) = fda L2 dx L2 (1 ± x), and L ̧(1±x) = ƒa ̧.* and so on in succession. Generally, the student will have no difficulty in demonstrating the following propositions : L1 (1 − x2) = 2′′ −1 L„ (1 + x) + 2′′−1 L„(1 − x), L2n(1 + x) + Lan (1 + 2) = 2 Lan(2) + 2 L2n−2 (2) 2 20 L2n−1(1 + x) − L2n-1 + 2 L2n-4(2) (log x) 1.2.3 x = (log x)2 1.2 (log x)2" 1.2.3.4...2n' 2 Lan-2 (2) log x Spence has extended this analysis to the investigation of the properties of transcendents defined by the general law the final function or (a) being such that it remains un changed when is substituted for a, or But for this investigation and others connected with it the reader is referred to the work before quoted. The transcendents which we have been considering are all such that they may be derived by direct integration from known functions, but there are many other transcendents which are given only by means of differential equations. As these are frequently functions of great utility in physical researches, the study of their properties without integrating the equations in which they are involved becomes of great importance. Two examples of such investigations are subjoined. (10) Let V be a function of a and r given by the differential equation of the second order d dx in which g, k, and are functions of r, and r is a variable parameter; and if V also satisfy the conditions From the given equation (1) we easily obtain therefore integrating with respect to x, (†m − rn) fdæg Vm Vn · k ( Vm = dV da V v. dr.). ; But from the condition (2) we find on taking the limit x = x1, that Similarly we find from (3) that at the limit 1 = x, the same relation holds; hence that As r„ and r„ are supposed not to be the same, it follows as may be deduced by the usual method for evaluating indeterminate functions. It is to be observed that the equation (3) involves an equation to determine r, which equation may be written as F(r) = 0. Poisson has shewn that this equation has an infinite number of real and unequal roots, for the demonstration of which proposition I must refer to the works cited below. • Bulletin de la Société Philomatique, 1828. Théorie de la Chaleur, p. 178. The function is of great importance in the theory of heat, and the investigation of its properties has formed the subject of several elaborate memoirs by MM. Sturm and Liouville. See Journal de Mathématiques, Tome 1. pages 106, 253, 269, 373, and Tome II. p. 16. m (11) Let Y and Z be integral and rational functions of μ, (1-μ3)3, cos w and sino determined by the equations A [ du 2 dw Ym Z1 = 0, so long as m and n are different. Multiply both equations by (1 μ3), and assume Multiply (1) by Z,dt dw and (2) by Ymdt dw, subtract (2) from (1) and integrate with respect to t and w. Then transposing, and observing that (1 − μ3) dt = du, we have Now if we effect the integration of the first term of the right hand side with respect to t, it becomes |