furnishes the values of the quantities P, Q, R, S, &c. In fact, the value of the sine being expressed by the equation 213. Suppose now that P, Q, R, S, &c., represent the sums of the different products which can be made with the 1 1 1 1 1 1 n fractions &c., from which the fraction 12' 2' 32' 42' 52' has been removed, n being any integer whatever; it is required to determine P, Q, R, S, &c., by means of P, Q, R, S, &c. If we denote by the products of the factors (1 − 1) (1 − 2) (1 − 3) (1 − 1) &c., F. H. 12 among which the factor (1 2only has been omitted; it follows that on multiplying by (1 − 3) the quantity we obtain or Employing the known values of P, Q, R, S, and making n equal to 1, 2, 3, 4, 5, &c. successively, we shall have the values of PQ,R,S,, &c.; those of PQ,R,S,, &c.; those of PQR ̧§ ̧, &c. 2 2 214. From the foregoing theory it follows that the values of a, b, c, d, e, &c., derived from the equations a+2b+3c + 4d + 5e + &c. = A, a+2b+3°c + 43d + 53e + &c. = B, a + 2b+3°c + 43d + 53e + &c. = C', &c., 215. Knowing the values of a, b, c, d, e, &c., we can substitute them in the proposed equation (x) = a sin x+b sin 2x + c sin 3x + d sin 4x + e sin 5x + &c., and writing also instead of the quantities A, B, C, D, E, &c., their values p' (0), "" (0), $'(0), ø11(0), 4" (0), &c., we have the general We may make use of the preceding series to reduce into a series of sines of multiple arcs any proposed function whose development contains only odd powers of the variable. 216. The first case which presents itself is that in which (x) = x; we find then '(0) = 1, 4′′ (0) = 0, $*(0) = 0, &c., and so for the rest. We have therefore the series which has been given by Euler. If we suppose the proposed function to be a3, we shall have $'(0) = 0, 4""'(0) = [3, '(0) = 0, $TM" (0) = 0, &c., which gives the equation We should arrive at the same result, starting from the preceding equation, In fact, multiplying each member by dx, and integrating, we If now we write instead of x its value derived from the we shall obtain the same equation as above, namely, We could arrive in the same manner at the development in series of multiple arcs of the powers x3, x2, xo, &c., and in general every function whose development contains only odd powers of the variable. 217. Equation (A), (Art. 216), can be put under a simpler form, which we may now indicate. We remark first, that part of the coefficient of sin x is the series $'(0) + 3 4" (0) + $ (0) + が ¿vii (0) + &c., |