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. Prop. To prove that a function of u and w can be arranged in only one series of Laplace's Functions. 34. For if possible let both these be true, F(x, w) = F,+F, +F, + ...... +F,+ ...... F(u, w) = G,+Gg+G,+ ...... + G + ...... :. 0=(F. - G.) + (F; - G) + ... + (F:- G.) + ...... and if these letters be accented when u' and w' are the variables instead of u and w, then 0 = (F) - G.') + (F - G,') + ...... + (F - G') + ...... .. 0= SSP.(F– 6.) dj' da, by Art. 26 But the principle demonstrated in the last Proposition shows that F.- G.= 1.5*(1+3P, + ...) (F) – G') dy'do', 2071 SP.(F; – G) dy'do", by Art. 26, = 0, by the condition deduced above; 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, Pi... 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 [1 +6* – 2c {' + W1-* V1 – re" cos (w – w')}], and is therefore a rational and entire function of Me V1-up cos w, and V1-sin w; and is precisely the same function of j's N1 - u cos w', and V1 - pe" sin w'. The general term of Pi, viz. that involving cos n(w-w'), arise solely from the powers n, n + 2, n +4, ... of cos w -w'). Hence (1 – 12) will occur as a factor of that term : and the other part of its coefficient will be a factor of the form Auin + Aunt + Aguin 28 + H, suppose. Hence P=H.+(1-2)* H, cos (w-w')+...+(1-u?)? E 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 can n ni-n-28 to As dition from which to calculate the arbitrary constants we have introduced. This condition, after reduction and arrangement, is as follows: d 0=(i – n) (i+n+1) H, (1 – 2)" + du dus Substituting in this the series which H, represents, and equating the coefficient of the general term (1 – u?)"pe zero, and reducing, we arrive at the formula (i-n – 2s + 2) (i – n— 28 +1) 4, 28 (21 – 2s +1) By making s successively equal 1, 2, 3 ... We have A,A,... in terms of A. Let these be substituted, and we have the coefficient of cos n (w – w' = (i – n) (i — n-1) 2 (21-1) call this Af(u). The coefficient A, is a function of pe', but is independent of u: and because P, is the same function of ré that it is of My it follows that A. =anf (u'), where an is a numerical quantity: and the coefficient of cos n (w – w') = Anf (u)f (u). To find an we must compare the first term of the ascending expansion of an f (u)f(u) in powers of u with the corresponding term in the coefficient of c' in the actual expansion of [1+0 – 2c {upe ' + V1 – pe V1 – pe" cos (w – w')}]". This leads to the following result: 1.3.5... (2i – 1)2 i (i – 1)... i - n +1) 1. 2.3... (i + 1) (2+2) ... (i+n) this applies when n=1, 2, 3 ..., but evidently not when 0: &, is found by equating coefficients to be (1.2.3... (2i - 1922 1.2.3... an = 2 na P,= We have now the complete value of P, in a series; it is as follows: $1.3.5... ? + M &c. 2 (2i – 1) 2 (2i – 1) 4 (2i – 3) ili -1) u"* + ) — u" - &c. 2 (21-1) 2 (21-1) 4 (2i – 3) x i +2 cos (W -w') i +1 (2-1)(3-2) Х M3+ (i-1)(1-2) (0-3)(8-4) x u" + i(i-1) +2 cos2(w-w') (i+1)(2+2) + (2-2)(2-3) (3-2)(2-3) (-4)(2-3) x M" + *(1-x) feat 1 + &...] 38. The following numerical examples are written down for convenience of reference : (1) P= peu' +V1 - MeV1 – ha cos (w – w'). 39. The following are some examples of expanding a function in a series of Laplace's Functions, by an application of the formula 2i+11 Fi F(r, w') Pidu'dw', 47 proved in Art. 32. Ex. 1. Arrange a + bu? in terms of Laplace’s Functions. Here F (u', w') = a + bu's. First put i =0, P = 1; i. Fo F.= #. $*(a + byen) dy' de (a+bu) dpi =-(ax' + bu" + const.) = a + 6. Again, put i=1, P, is found in the last Article. between the proper limits, w'= 0 and w'= 27, between the limits u'=-1 and j'= 1, = 0. P. A. 3 |