π 15. The value of the definite integral (3log (1 + n cos3 0) d☺ 0 may be found whatever positive value is given to n from the formula π π log (1+n cos20) d0 = log {(1+n) (1+n,)*(1+n ̧)*...........} where n, n,, n...... are quantities connected by the equation π Nr+1 = 2 (Put 0=—— 0′; see Art. 41.) 16. Shew that where tan 6=2. Hence shew that if e cos ax be α с 64 CHAPTER V. DOUBLE INTEGRATION. 56. LET (x) denote any function of x; then we have seen that the integral of (x) is a quantity u such that (x). The integral may also be regarded as the limit du dx = of a certain sum (see Arts. 2-6), and hence is derived the symbol (p(a) da by which the integral is denoted. We now proceed to extend these conceptions of an integral to cases where we have more than one independent variable. 57. Suppose we have to find the value of u which satis fies the equation d2u dy dx = (x, y), where p(x, y) is a function of the independent variables x and y. The equation may be written or du if v = Thus v must be a function such that if we differ entiate it with respect to y, considering x as constant, the result will be (x, y). We may therefore put Hence u must be such a function that if we differentiate it with respect to x, considering y constant, the result will be the function denoted by (x, y) dy. Hence The method of obtaining u may be described by saying that we first integrate (x, y) with respect to y, and then integrate the result with respect to x. The above expression for u may be more concisely written thus dx [f(x, y) dy de, or f(x, y) da dy. On this point of notation writers are not quite uniform; we shall in the present work adopt the latter form, that is, of the two symbols dx and dy we shall put dy to the right, when we consider the integration with respect to y performed before the integration with respect to x, and vice versa. 58. We might find u by integrating first with respect to x and then with respect to y; this process would be indicated by the equation u = [[p(x, y) dy dx. 59. Since we have thus two methods of finding u from the d'u equation = (x, y), it will be desirable to investigate if dx dy more than one result can be obtained. Suppose then that Ալ and u are two functions either of which when put for u satisfies the given equation, so that dw dx = Now from an equation =0 we infer that w must be a in constant, that is, must be a constant so far as relates to x; other words, w cannot be a function of X, but may be a function of any other variable which occurs in the question we are considering. Here the constant, as we call it, must not contain y, but may contain x; we may denote it by x(x). And [f(y) dy we will denote by (y); thus finally d'u v=¥ (y) +x (x). Therefore two values of u which satisfy the equation (x, y) can only differ by the sum of two arbitrary functions, one of x only and the other of y only. dx dy = 60. We shall now shew the connexion between double integration and summation. Let (x, y) be a function of x and y, which remains finite and continuous so long as x lies between the fixed values a and b, and y between the fixed values a and B. Let a, x, xb be a series of quantities in order of magnitude; also let a, y, Y. • • • • Ym - 1, B be another series of quantities in order of magnitude. Let also let 19 .... We propose now to find the limit of the sum of a certain series which consists of every term of the form hçk ̧ † (X,_11 Ys_1), where r takes all integral values between 1 and n, and s takes all integral values between 1 and m; also x and y. Уо are to be considered equivalent to a and a respectively. Thus we may take hko (x, y) as the type of the terms we wish to sum, or we may take Ax Ay (x, y) as a still more expressive symbol. The series then is h ̧{k ̧¢ (a, a)+k, $(a, y1) +k ̧$(a, y2)..............+km&(a, Ym-1)} + h2 {k ̧¤ (x ̧, α) + k ̧¢ (x ̧‚ Y1) +k ̧Þ (X ̧‚Y2) ..............+km (x12, Ym-1)} 19 a) 2 19 + h‚ {k ̧ † (xn−19 α) + k2 $ (Xn-1 Y1) + kg $ (Xn-19 Y2).................. 19 + km & (Xn-12 Ym_1)}• Consider one of the horizontal rows of terms which we may write hr+1 {k, & (x,, α) + k2Þ (Xr, Y1) + k ̧ $ (Xr, Y2) ................+ km $ (X,, Ym_1)}• 2 The limit of the series within the brackets when k1, k„,... km are indefinitely diminished is, by Art. 3, Since this is the limit of the series, we may suppose the series itself equal to $ (x, y) dy +Pr+1? a where Pr1 ultimately vanishes. Pr+1 B Let (*$ (x, y) dy be denoted by ✨ (~,); then add all the a horizontal rows and we obtain a result which we may denote by Σh f (x) + Σhp. |