dition from which to calculate the arbitrary constants we have introduced. This condition, after reduction and arrangement, is as follows: ni-n-28 to Substituting in this the series which H, represents, and equating the coefficient of the general term (1— μ2)"μ" zero, and reducing, we arrive at the formula By making s successively equal 1, 2, 3 ... we have 4,4,... in terms of A. Let these be substituted, and we have the coefficient of cos n (w — w' = call this Af(μ). The coefficient A is a function of μ', but is independent of μ: and because P is the same function of μ' that it is of μ, it follows that A=anf(u), where a, is a numerical quantity: and the coefficient of an cos n (w-w') = anf(u') ƒ (μ). To find a, we must compare the first term of the ascending expansion of anf (u')f(u) in powers of μ with the corresponding term in the coefficient of c' in the actual expansion of 12 [1+c2−2c {μμ' +√1 − μ2 √1 − μ22 cos (w — w')}] ̄*. This leads to the following result: (1 . 3 . 5 ... (2¿ — 1))2 ¿ (¿ — 1) ... (i − n + 1) n= this applies when n = 1, 2, 3..., but evidently not when n=0: a, is found by equating coefficients to be (1.2.3... (2-1))2 1.2.3...i We have now the complete value of P, in a series; it is as follows: 38. The following numerical examples are written down for convenience of reference : 39. The following are some examples of expanding a function in a series of Laplace's Functions, by an application of the formula Ex. 1. Arrange a + bμ2 in terms of Laplace's Functions. Here Fu', w') = a+bμ2. First put i=0, P=1; Again, put i=1, P, is found in the last Article. 1 -1 0 1 4π -1 between the proper limits, w' = 0 and w' = 2π, 3 between the limits μ': · 1 and μ' = 1, = 0. P. A. 34 Next, put = 2, and substitute for P, from the last Article. 2 Hence the function a+bu2 stands as follows, when arranged in terms of Laplace's Functions, (a + b) + b (μ3 — — ), and consists of two Functions, of the order 0 and 2 respectively. The above is a long process to arrive at this result. It might have been so arranged at a glance. But the calculation has been given as an example of the use of the formula, which in most cases is the only means of obtaining the desired result. 2 Ex. 2. Arrange 49+30μ+3μ3+ √1−μ3 (40+72μ) cos (w—a) +24 (1 − μ3) cos 2 (wa) in terms of Laplace's Functions. 2 The result is 50+ {30μ + 40 √√1 − μ2 cos (w — a)} + {3μ2 − 1 + 72μ √1 — μ3 cos (∞ − a) +24 (1 − μ2) cos 2 (w—a)}, consisting of three functions of the orders 0, 1, 2. Ex. 3. Let the function be 1+√√2 − 2μ3 cos (w + a) + — (1 − μ2) cos 2 (w+a). |