24. COR. The same equation is true for V, for an external particle. If the particle be part of the attracting mass, the second side will be - 4πp' instead of zero. PROP. To explain the method of expanding R in a series. 25. The expression for R becomes, when the polar co-ordinates are substituted, 12 12 [x2 + r'2 — 2rr' {μμ' + √1 − μ3 √1 — μa2 cos (w — w')}] ̄*, and this may be expanded into either of the series where P, P, ...P... are all determinate rational and entire functions of μg The general coefficient P; is of i dimensions in ' The greatest value of P; (disregarding its sign) is unity. For if we put 12 μμ' + √1 − μ2 √ 1 — μ”2 cos (∞ — w') = cos p = ↓ (z + 1), μμ (w (1 + c2 — 2c cos $)3, or (1 — cz)-1 coefficient of c2 in or (1 − cz)-1 (1 − 2) * A 1+ 1 c 1 = 1 ( + 1 ) + B (b2 + 1 ) + .. = 24 cos ip+2Bcos (-2)+... A, B... being all positive and finite. The greatest value of this is, when = 0. Hence P; is greatest when p = 0. But then P= coefficient of c' in (1+c2 - 2c) or (1 — c)1 coefficient of c in 1+c+c2 + ... + c2+... = Hence 1 is the greatest value of P. It follows that the first or second of series (1) will be convergent according as r is less than or greater r'. To obtain equations for calculating the coefficients P, P1, P... substitute either of the series (1) in the differential equation in R in the last article, and equate the coefficients of the several powers of r to zero. The general term gives the following equation: by integrating which P; should be determined*. The series for R would then be known. For the direct integration of this equation, see two Papers in the Philosophical Transactions for 1841 and 1857, by Mr Hargreave and Professor Donkin respectively. LAPLACE'S COEFFICIENTS AND EQUATION. 21 26. The functions P, P...P... possess some remarkable properties which were discovered by Laplace. They are therefore called, after him, Laplace's Coefficients, of the orders 0, 1,...... It will be observed that these quantities are definite and have no arbitrary constants in them. Laplace's Coefficients are therefore certain definite expressions involving only numerical quantities with and w, u' and w'. Any other expressions which may satisfy the partial differential equation in P, which is called Laplace's Equation, may be designated Laplace's Functions to distinguish them from the "Coefficients." The fundamental properties of these Coefficients and Functions we shall now proceed to demonstrate. μ PROP. To prove that if Qi and R, be two Laplace's Co 1 2π efficients or Functions, then [*[** QR, dμ dw = 0, when i and d Qi since when = 0 and 27, each of the functions Q, R, has the same values, because they are functions of μ, 1 2π (1 − μ2) 1 2π dR 1 d R + άμ -1 0 Hence, [[ QR, dude =0, when i and ï are unequal. -1 0 When they are the same the equation becomes an identical one, and therefore gives no result. This property is true also when i=0, as may easily be shown by going through the process of the last Proposition, Q being Q, or a constant. PROP. To prove that a function of p, √1-μ2 cos w, and √1-μ2 sin w, as F (μ, w), can be expanded in a series of Laplace's Functions; provided that F(μ, w) do not become infinite between the limits - 1 and 1 of μ, and 0 and 2 of w. 28. This very important Proposition will occupy the present and four following Articles. Let μμ'+√1-μ2 √1-μ" cos (ww') =p: then by Art. 25, (1 + c2 — 2cp) ̄* = 1 + P ̧c + P2c2 + ...... P¿¢1 + ...... EXPANSION IN TERMS OF LAPLACE'S FUNCTIONS. Differentiate with respect to c, p-c (1 + c2 − 2cp) } = P1 + 2P2c+ 23 Multiply this by 2c and add to it the former equation; 1-c2 (1 + c2 — 2cp) # =1+3P1c+5P ̧c2 + ...... + (2i + 1) P1c; + ... Now c being quite arbitrary we may put it = 1. Then the fraction on the left-hand side of this equation vanishes, except when p = 1; in which case the fraction on the left hand becomes apparently indeterminate: but it is in reality infinite. 12 and that this may not be greater than unity we must take μ2+μ22 not greater than 2uu, or (μ~μ) not greater than zero. Hence μ = μ, and therefore cos (w' - w) = 1, and w'=w. 1 2 Hence, the series 1+ 3P, + 5P+......+ (2i+1) P; +...... vanishes for all values of μ and w, u' and ', except when μ = μ' and w = w', in which case the sum of its terms suddenly changes from zero to infinity. 29. Upon this series depends the important property of Laplace's Functions which we are now demonstrating, and which gives them so great a value in the higher branches of analysis. Our demonstration consists in showing, that 1 2π =4π F(μ, w) when c = 1, Aπ F' (μ, ∞) = ['* [*” ̃(1 + 3P, +5P ̧+.....) F' (μ', w') dμ' dœ'; -1~0 from which property, as will be seen in the end, our Proposition, as enunciated, immediately flows. |