Cauchy, Mémoires des Savans Etrangers, Vol. 1. p. 638. (10) da a2±2 Calling the definite integral u, and differentiating with respect to r, we have On integrating the second side with respect to x by parts, the equation becomes since the integrated part vanishes at both limits, and the unintegrated part when taken between the limits is equal to u. This equation on integration gives π Laplace, Mémoires de l'Institut, 1810, p. 290. To determine the constants, we observe that u cannot increase continually with a, and therefore the term involving e must vanish, or C = 0. This being the case we have, when a = 0, On differentiating this with respect to a, there results Integrating with respect to a and determining the constant so as to make the integral vanish with a, we find It is to be observed that the formula (a) is discontinuous, as the integral is equal to " when a is negative. ΠΕ -a re" when a is positive, and to Libri* has accordingly expressed the value of the integral in the following manner: Laplace, Mémoires de l'Institut, 1810, p. 295. (a) S Cah. xv1. p. 225 (Poisson), and for that of "dx 0 cos ax (1 + x2)2π see Jour. de Mathématiques, Vol. v. p. 110 (Catalan). 1 If we expand the denominator we have the series. a3 (1 + 2a cos x + 2 a3 cos 2x + &c. + 2 a' cos rx + &c.) Multiply by da cos ra and integrate: every term vanishes at both limits except 2 a′ f* dx (cos rx)2= a' f′′ dx (1 + cos 2rx) = πa'; therefore the reader may consult Legendre, Exercices, Vol. 1. p. 373. * Crelle's Journal, Vol. x. p. 309. By a similar expansion it is easily seen that (b) da log (1 − 2a cos x + a2) = 0, according as a is less or greater than 1. or 2π log a, Poisson, Journal de l'Ecole Polytechnique, Cah. xvII. p. 617. Also in like manner we find (c) fo da cos r x log (1 − 2 a cos x+a2) according as a < or > 1. Expand the second factor as before, and integrate each term separately by (11, a); then on summing the result, we 8 dx = 2 π 6 -a (d) log (sin)-log(") 1 + x* = 2 2 Changing the sign of a and then making a = 1, we have 28 dx () log (tan) = log() 1+x 2 Changing the sign of a and then making a = 1, we have The formula (a), (b), (c) are due to Legendre: see his Exercices, Vol. 11. p. 123. The formula (d), (e), (f), (g), (h) were first given by Georges Bidone, in the Mémoires de Turin, Vol. xx. (14) To find the value of da log (sin x) = f da log (cos x). By Cotes's Theorem we have |