+ = 0; r dr dR) Hence the equation in R becomes 1 d'R + + * sin Ꮎ dᎾ pa sino O dw? d?.rR d'R cos Ꮎ dᎡ 1 d’R + 0. do sin Ꮎ dᎾ sina A dw? Put cos 0 M, then 1 d'R + = 0. dus *1-* dwa PROP. To explain the method of expanding R in a series. 24. The expression for R becomes, when the polar co-ordinates are substituted, [m2 + go— 2rr' {up' + VI - Vi pe" cos (w – w')}]"}, and this may be expanded into either of the series P. + P +... P + ... (1), 1 z' + P, +P; t... piti piti where P., P , ... P.;... are all determinate rational and entire functions of us N1 -cosw, and Wi-u' sin w; and the same functions of u', N1-u' cos w', and V1 – re" sin w'. The general coefficient P, is of i dimensions in N1 - Ma cos w, and N1 p? sin w. The greatest value of P; (disregarding its sign) is unity. For if we put then P = coefficient of d in (1 +0° – 2c cos 6)**, or (1 – cz)-1 (1-3)* =2A cos id + 2B cos (i — 2)$+... A, B... being all positive and finite. The greatest value of this is; when p=0. Hence P, is greatest when = 0. But then P:= coefficient of c' in (1 + 0* — 2c) or (1 — c)-2 = coefficient of d' in 1+0+0+ ... + 0 + ... = 1. 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 than r'. To obtain equations for calculating the coefficients P, P, ... 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 : d dP! đP; + 1 by integrating which P, should be determined*. The series for R would then be known. 1 25. The functions Po, 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, ... i ... It will be observed that these quantities are definite * 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. and have no arbitrary constants in them. Laplace’s Coefficients are therefore certain definite expressions involving only numerical quantities with u 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. Prop. To prove that if Q, and R, be two Laplace's Coefficients or Functions, then when i and i are different integers. 26. By Laplace's Equation in the last Article d dQ 1 d'Q. ili+1) Qi=- (1 d die du-1--* do; Q,du dQ since when w= 0 and 27, each of the functions Q., Ri', do dR' has the same values, because they are functions of , do N1 - po cos w and V1 – Må sin w. Hence, | S* Q,Rdu do 1 d dR , (1-1) + ilė + 1) 1 - dwa i ' (' + 1) ili+) by Laplace's Equation. Hence, | S* Q, Rydudw=0, when i and i' are unequal. 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, Vi being Q. or a constant. PROP. To prove that a function of M, N 1 – je? cos w, and N1 - r* sin w, as F (u, w), can be expanded in a series of Laplace's Functions, provided that Fly, w) do not become infinite between the limits – 1 and 1 of , and 0 and 27 of w. 27. This very important Proposition will occupy the present and five following Articles. Let up' +V1 – je? V1 – r'a cos (w – w) =p; then by Art. 24, . (1+0—2cp)+=1+Pc+Pc + ...... +Pc + ...... c being any quantity not greater than unity. Differentiate with respect to c, P-C P, +2Pc + ...... + iP.c4 + ...... .. Multiply this by 2c and add it to the former equation. 1 -ca =1+3P,c +5P,c*+ (1 +0% – 2cp) ...... +(2i+1)P.Ci + ... 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. 1-c2 1+c For when p=1, (1+0 – 2cp)*(1 -c)2 infinity, when c=1. When p=1, then 1-uu 1-2μμ' + μμ” cos (w' – w) = V(1—?) (1 – Me") -M2 M2 tu2u2 and that this may not be greater than unity we must take + r not greater than 2pin', or (u ~ p')not greater than zero. Hence u' = j, and therefore cos (w'-w) = 1, and w'=w These, then, are the values of u' and w' which make p=1. Hence, the series 1 +3P, +5P, + + 2i +1) Pi +...... vanishes for all values of u and w, r' and w', except when u=j' and w=w', in which case the sum of its terms suddenly changes from zero to infinity. ,2,12 > 28. 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. In consequence of the discontinuity above pointed out, and also because the series becomes infinite in one stage of the variations of its variables, it has been considered by some to be unsatisfactory to deduce any properties from it. But the latter objection is entirely removed by the fact, that we do not use the series in its present form, but after being multiplied by small infinitesimal quantities which render the aggregate of its terms finite,' preventing their accumulating to an infinite amount. With regard to the objection of discontinuity, there appears to be no sufficient ground for it. There is no question, that the property deduced (as enunciated |