cos Y if m is even. sin % m-2 sin z + sin z + is any even number 2k, then y cost y_y (1-siny), an expression (m-2) (-4)...1 I tan 4 y, if m is odd, and to (m-1) (m--3)...1 sin y 93. Suppose y=90°—, and the preceding formula will give 2 1 s + cos" % -1 cos" z' (m-1) (m—3) cos” (m—2) (m-4) + &c....to the term (m-1)(m-3) (m-5) cos' (m-2) (-4)...2 sin z if m is even, and to the term (m-1) (m -3)...1 (m-2) (M-2)...1 2 1 tang (45°+ z), (m-1) (m—3)...2 (m-1) ( m3)...2 5 5.3.1 s + Itan (45° +ày). 6.4 cosy' 6.4.2 cos y 6.4.2 Hence it is easy to integrate the formula y cosa y, for if m is an 2k +1 odd number, as 2k+1, we have y cos y_flux (sin y) (1-sin? g), which is evidently integrable for any value of n. sin" y COS Z m + COS 2 gin' y + sin y sin y + coso y which, when developed. is easily integrated by the formula for sy cos" y 95. The same process will also apply to y sin" y; and the formula y may be integrated upon similar principles ; so that it is sin' y cosa y easy to integrate any fluxional expression containing sines and cosines, provided that they are susceptible of integration. Examples. 1. Required the fluent of 2 sin. z cosz? 2. sino x cos z ? ON THE INTEGRATION OF FLUXIONS CONTAINING SEVERAL VARIABLE QUANTITIES. 96. In any function t, of two variables x and y, if we first take the fluxion upon the supposition that x alone is variable, every term not originally containing some power of r will disappear, and the result will be of the form Pr. If we then take the fluxion of Pi, upon the supposition that y alone is variable, the result will be of the form P'xy, and by the double operation, all terms not originally containing both x and y, will have disappeared. Again, if we take the fluxion of the same function i, upon the supposition that y alone is variable, the result will be of the form Qy; and if we then take the fuxion of Qý, upon the supposition that I alone is variable, the result will be of the form Q'yx, all terms being excluded here also which did not originally contain some powers of both x and y. Now in any term including powers of both x and y, we shall obviously obtain the same result, whether we take its fluxion, first supposing only x to vary, and then take the fluxion of the result, supposing y alone vary; or whether we first supposey alone to vary, and then r. Hence we shall always have Pxy=Q'yx, or more simply P'=Q. The quantities Pä, Qy are called the partial fluxions of the function t, the first taken relative to x, the second relative to y; and P Q are called the parlial fluxional coefficients. Example. Let the function be z* + x3y +a? y’. Then the partial fluxion relative x is (4x3 + 3.x? y) x; and the partial fluxion of (4.x} +3.22 y) taken relative to y, is 3.x2 xy. Again the partial fluxion of 2* + xy + aạy relative to y is (x3 +2 a’ y) y, and the fiuxion of (x3 +2a’ y) y, supposing x only to be variable, is 3.x? y x, the same result as before. The partial fluxion of the function t;, taken relative to x, that is i supposing x alone to vary, is usually expressed by m; and the par to tial Auxion relative to y, by Ly; the denominators shewing the y variable relative to which the fluxion is taken. Consequently the entire fluxion of any function t, of two variable it and y, may be represented by + ;; where! i are the partial fluxional y y coefficients of the first order, and correspond to P and Q in the preceding theorem. i Upon the same principles of notation, the partial fluxion of 3 i. · y, taken relative to y, will be expressed by x or more simply by y 17 y; and the partial fluxion of relative to x, by ; x: where ху y yu 7 and correspond to P and Q' of the preceding example, and ху ух are called partial fluxional coefficients of the second order ; the denominator xy denoting that the fluxion is first taken upon the supposition of x alone varying, and then again of y alone varying; and the denominator yr denoting an arrangement exactly the reverse. 97. From the preceding theorem it follows therefore that, the flucional coefficient of the second order of any function of two variables, taken first with respect to one of these variables, and then with respect lo the other, is the same in ‘whatever order the operations are performed. And consequently, if any fluxional expression of the form Px +Qy admits of exact integration, the fluxional coefficient of Q taken relative to x, must be equal to the fluxional coefficient of P, taken relative to y. 98. If this condition is fulfilled, the integration is easy. For since__ =Pë, if we take the fluents, supposing only z as variable, we shall have t=S Péta correction which may be some function Y of y, as is evident. Therefore t or S (Px+Qy)=Pc+Y. Similarly, integrating on the supposition of y, being the only variable, we have 1=S (Px+Qy)=S Qý +a function X of x. Therefore SQy+X=í Pi+Y, or by transposition, SQy-Pr=Y-X. 2 |