Then by mere expansion, omitting the squares and higher powers of h as they vanish in the limit with reference to the first power, we see the truth of the following; 1 - 1 1 {√ 2) '(1-c)2+2cxh √(1-c)2+2c (x+1) By giving x its successive values from 0 to n-1, and adding together all the resulting values of this expression and taking the limit, we have the integral with respect to p. It matters not in which order we effect the integration. Hence the whole integral Here it will be seen that the terms within the last brackets mutually destroy each other whatever be the value of c. It may also be observed that were this not the case, that whole part of the expression would vanish for the particular value c=1 (which is the only case we shall have to use), whatever the value of the sum of the terms following the multiplier 1- c2, so long as that sum is not infinite. 32. Suppose now Fu', w') is any function of μ' and w', then 2π = √ √ * (1 + 3 P, + ... + (2i + 1)“P, + ...} F' (μ', w') dμ'dw'. -1 0 The reasoning above enables us now to prove that the integral on the left-hand side = 4πF (μ, w), which directly leads to the theorem we are wishing to demonstrate. The function F(u', w') at the point Q is F(u, w), call it F: let F", F"... F') be its values at the points of junction of the successive elements along the great circle QPq. Then by 1 1 1 multiplying the successive values of (42) by F, F', с F"... and adding them together, we have value, being the ratios of QD to the successive values of DP. When c=1 each of them vanishes; and in the limit none of the factors F" — F, F" — F', ... become infinite. Hence the integral = ["dy. 2.F, when c=1, = 4πF (μ, w) because F(μ, ∞) is a function of μ and w only, and is altogether independent of y. 1 2π .• . 4πF (μ, w) = ['* * * {1+3P, +...+(2i+1)P‚+........}F'(μ'‚w')dμ'do'; -1 0 which we will call F, is a function of μ and w; and evidently satisfies Laplace's Equation in μ and w, because P, does so. Hence, this is a Laplace's Function, of the ith order: and the result is, what we were to demonstrate, that any function of μ and o may be expanded in a series of Laplace's Functions; м or, 2 33. Those who are at all acquainted with the controversy which followed the first discovery of these remarkable functions by Laplace, will understand why we have entered so fully upon the subject. Laplace's demonstration in the Mécanique Céleste was by no means conclusive. This Mr Ivory pointed out in the Philosophical Transactions for 1812; and in the Volume for 1822 he threw considerable doubt upon the applicability of the theorem to functions that are not rational and entire functions of μ, V1-μ cos w, √1-μ sin w. Poisson has written much upon the subject. In the first edition of the author's Mechanical Philosophy the last method of Poisson was followed, as given in his Théorie Mathématique de la Chaleur; in which he effects the integration of the fraction on the left-hand side by the artifice of substituting for it an integrable, but entirely different fraction in its general form, but which coincides with it in the particular case for which he requires it in the result, viz. when c=1. In the Second μη Edition of the Mechanical Philosophy we gave a much shorter proof, based upon an idea taken from Professor O'Brien's Mathematical Tracts. But this also rather concealed the real difficulty of the case, and passed it over by an artifice. In the demonstration now given, we have gone to the foundation of the calculus, the doctrine of limits, and attempted to clear up all difficulty and ambiguity in the matter. With regard to the doubt thrown out by Ivory, alluded to above, it seems to be clear that theoretically every function can be expanded in a series of Laplace's Functions: but if it be not a rational function of the co-ordinates, the number of terms in the series will be infinite, and if the terms be not convergent, the expansion, or rather arrangement, will be useless. But this must be determined in each case. A similar uncertainty, requiring examination, always attends the use of infinite series. μ PROP. To prove that a function of μ and w can be arranged in only one series of Laplace's Functions. 34. For if possible let both these be true, 1 2 F(μ, w) = F + F12+ F2+......+F+ ........ 1 2 .. 0 = (F. — G。) + (F1 − G1) + ...... + (F; — G1) + ...... and if these letters be accented when μ' and w' are the variables instead of μ and w, then 0 = (F' — G.') + (F,' — G1') + ...... + (F! — G2) + ...... .. 0 = 1 2π - [*[* P. (F! – G¦') dμ' do', by Art. 26. = 0 But the principle demonstrated in the last Proposition shows that therefore F= G., and the two series are term by term identical, and the Proposition is true. 35. It follows from this, that if by any process we can expand a function in a series of quantities which satisfy Laplace's Equation, that is the only series of the kind into which it can be expanded: and if by any other process we obtain what is apparently another, the terms of the two series must be the same, term by term, and we may put them equal to each other. 36. Before concluding this Chapter, we shall explain how the numerical coefficients in PP... P... are found: and shall give a few examples of the truth of the last Proposition but one (that in Art. 32) by actual integration. PROP. To explain how to expand P. 37. By Art. 24 P is the coefficient of c in the expansion of the function 12 [1 + c2 − 2c {μμ' +√1 − μ2 √1 −μ”2 cos (w — w')}] ̄*, and is therefore a rational and entire function of μ, 2 √1-μ2 cos w, and √1-μ2 sin w; and is precisely the same function of μ', 12 12 √1-μ cos w', and √1 — μ'2 sin w'. The general term of P, viz. that involving cos n (w-w'), can arise solely from the powers n, n + 2, n + 4, ... of cos (w — w'). n - Hence (1-2) will occur as a factor of that term: and the other part of its coefficient will be a factor of the form P ̧=H ̧+ (1 − μ2)1 H ̧ cos (w−w') +...+(1—μ3) H„cosn(w−w')+..... If this be substituted for P in Laplace's Equation and the coefficient of cos n (w - w') be equated to zero, we obtain a con |