tion partly because of its historical interest, and partly because it affords an instructive view of the subject. We shall however not stop to examine the demonstration closely, but proceed at once to the mode of investigation adopted by Poisson. 304. The following expansion may be obtained by ordinary Trigonometrical methods, h being less than unity, so that the series is convergent. Multiply both sides by (v), and integrate with respect to v between the limits - 7 and 7; also make h approach to unity as its limit. On the right-hand side the different powers of h become ultimately unity. The numerator of the fraction on the left-hand side will ultimately vanish, and thus the integral would vanish if the denominator of the fraction were But if x lies between 1 and 1, the term never zero. COS π (v-x) will become equal to unity during the integration, and thus the denominator of the fraction will be (1 − h)2, and will tend towards zero as h approaches unity. Thus the integral will not necessarily vanish; we proceed to ascertain its value. Let v-x=z and h=1-g, thus Now the only part of the integral which has any sensible value, is that which arises from very small positive or negative values of z; thus we may put Suppose a and -ẞ to be the limits of z; we thus get Hence, finally, when g is supposed to vanish, we have 2lp (x). Thus if x lies between 7 and 1, • (x) = 2/27 √ $ (v) dv + Σ If x does not lie between -7 and 7, the left-hand member must be replaced by zero. If however x = 7 or, then the integral on the left-hand side has its sensible part when v is indefinitely near to l and -1; we should then have to perform the above process in both cases, but the integral with respect to z would only extend in the former case from - B to 0, and in the latter from 0 to a. Hence instead of 21 (1) on the left-hand side, we should have 18 (1) + 1$ (− 1). 305. In the same way as the result in Art. 313 is found, we have, if we integrate between 0 and 1, 1 1 • (x) = {} [ ✨ (v) dv +} Σ;* [ $ (v) cos 0 0 this holds if x has any value between 0 and 7; but when x=0 the left-hand member must be (0), and when x=l the left-hand member must be (1); for all other values of x the left-hand member should be zero. this holds for any value of x between x=0 the left-hand member must be the left-hand member must be (1). This holds for any value of x between 0 and 7, both in This holds for any value of x between 0 and 7 both exclusive; and when x=0 or 7, the left-hand member should be zero. Equation (4) coincides with Lagrange's Formula. We will now give some examples. 306. Expand x in a series of sines. Take formula (4) of Art. 314, and suppose 7=π; then therefore v sin nv dv=if n be odd, and Thus n π if n be even. n x= 2 {sin x - sin 2x + sin 3x - sin 4x + ..............}. This holds for values of x between 0 and π, and as both sides vanish with x it holds when x=0; and it is obvious that if it holds for any positive value of x it holds for the corresponding negative value; hence it holds for values of between and π, exclusive of these limiting values. 307. Expand cos x in a series of sines. (4) of Art. 314 and suppose lπ; then = Take formula cos v sin nv dv = 4 [{sin (n + 1) v + sin (n − 1) v} dv (cos (n + 1) v cos (n - 1) + n-1 therefore 2 (4 sin 2x + sin 4x +...+ π 3 15 n2-1 This holds from x=0 to x=π, exclusive of these limiting values. 308. Expand x in a series of cosines. Take formula (3) of Art. 314, and suppose 7=π; then This holds from x=0 to x both inclusive. If we put x= -y, we obtain the following formula, π 1 52 cos 3x + cos 5x +......}. which holds for any value of y between — T π П and both in 2 2 Here 309. Expand ex-e-a in a series of sines. 310. Expand ea (≈ − x) + e−a (”—x) in a series of cosines. 311. We have shewn that the formula (3) of Art. 314 holds for any value of x between 0 and 7 both inclusive ; it is easy to determine what the right-hand member is equal to when x lies beyond these limits. Suppose a positive, and between 7 and 21; put x=21-x so that x' is less than 7, then therefore the value of the right-hand member is (a). Next suppose greater than 27; and suppose it equal to 2ml+x' where is less than 27; then |