+ (a2+n2) cos (nx+2() ax €a (x+h) cos n (x + h) = €a2 {cos nx + (a2 + n2)1 cos (nx + 4). h (r+h) cos® cos {(x+h) sin 0} = ecos {cos (v sine) + cos (æ sin0+0) tan-1a = (-)'-1. (r − 1) (r − 2) ... 2. 1 sin ry. (sin y)', therefore + sin 3y (sin y)3 - sin 4y (sin y)' + &c. From this development Euler* has deduced many remarkable theorems, some of which are subjoined. In the preceding example let h = -x, then 002 therefore tan-1 x = sin y. sin y. x + (sin y)2 sin 2y 2 = y + sin y cos y + sin 2y (cos y)2 + sin 3y (cos y)3 + &c. 0 = + cos y + cos 2y + cos 3y+ &c. In these formula y lies between 0 and T. Calc. Diff. p. 380. (7) Let u= cot-'a, then cot-x+h) is easily found from the expression for tan1 (x + h). For since and we have merely to substitute cot1 1 + a22 for tan-1a and to change the signs of the terms beginning with the second: and as in this case y = u, we find This Theorem, which is usually called Maclaurin's, but which ought to bear the name of Stirling, was first given by James Stirling in his Lineæ Tertii Ordinis Newtonianæ, p. 32. Maclaurin introduced it into his Treatise of Fluxions, p. 610, and his name has generally been given to the theorem from an erroneous idea that his work was the first in which it appeared. The following is the enunciation of the Theorem : If f(x) be a function of a, and if we represent the values. which it and its successive differential coefficients acquire when x = 0, by ƒ (0), ƒ'(0), ƒ" (0), ƒ" (0), &c.; then This Theorem is evidently a particular case of that of Taylor. Ex. (1) Let u = ƒ (x) = (1 + x)3 ; ƒ (0) = 1, |