and have no arbitrary constants in them. Laplace's Coefficients are therefore certain definite expressions involving only numerical quantities with μ and w, μ' 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 Co 1 2π efficients or Functions, then [["QR, dpdw = 0, when i and since when w = 0 and 2π, each of the functions Q, R, dR dw has the same values, because they are functions of μ, 1 2π dR + d Qi 1 -1 0 Hence, [*[*2. R. dude =0, when i and i' are unequal. -10 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 μ, √1 — μ3 cos w, and √1 — μ2 sin w, as F(u, w), can be expanded in a series of Laplace's Functions; provided that Fu, w) do not become infinite between the limits - 1 and 1 of μ, and 0 and 2π of w. 27. This very important Proposition will occupy the present and five following Articles. 2 12 Let μμ'+√1-μ3√1-μ cos (w-w') =p; then by Art. 24, (1+c2-2cp)=1+ P12c + P2c2 + ...... + P ̧c2 + . c being any quantity not greater than unity. Differentiate with respect to c, ...... Multiply this by 2c and add it to the former equation. 1-c2 (1 + c2 - 2cp)* =1+3Pc+5P2c2+...... + (2i+1) Pic; + ... 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. For when p=1, 1+c 1-c2 = 1 - μμ' and that this may not be greater than unity we must take μ2+μ not greater than 2up', or (u) not greater than zero. Hence μ' =μ, and therefore cos (ww) = 1, and w'=w. These, then, are the values of μ' and w' which make p = 1. 1 2 Hence, the series 1 + 3P1 + 5P2+ ...... + (2i + 1) P; +...... vanishes for all values of μ and w, u' and w', except when μ=μ' and w=w', in which case the sum of its terms suddenly changes from zero to infinity. 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 in our Proposition) is true, at any rate for rational functions of μ, √1-μ2 cos w, and √1-μ2 sin w, and is also most important. This objection, however, deserves to be examined with care, which we now propose to do in the course of our demonstration. 29. Multiply both sides of the last equation by the double element du'da', and integrate, (1-c2) dμ'dw' The property of Laplace's Functions proved in Art. 26, shows that every term of the series on the right, except the first, vanishes of itself, independently of the other terms; and therefore (as was before intimated) the terms cannot accumulate. The first term is 47: and therefore the integral of the fraction on the left, that is It is remarkable that this result is altogether independent of c. 30. The truth of this may be shown also by integrating the fraction on the left. This cannot readily be done with the co-ordinates as at present chosen. But it may be done by a simple transformation, and a change in the way of taking the elements. -1 Suppose a sphere of radius unity described about C the origin of co-ordinates. Let O' and w' be the angular co-ordinates to a point P, e' (or cosμ') measured from a fixed point A along a great circle of the sphere, and w' the angle which this great circle makes with another and fixed great circle through A. Then de'. dw' sin e', or du'do', is an infinitesimal element of the surface of the sphere at P. Take D a point within the sphere, and let CD=c, and sup in pose CD meets the sphere in when produced forwards, and 9 when produced backwards. Let μ and a be the co-ordinates of Q. Then p (see its value, Art. 27) is the cosine of the angle which CP and CQ make with each other: and the distance of P from D=V1+c-2cp. Let be the angle which the plane CPQ makes with CAQ, that is, the angle AQP. By changing the origin of the angles from A to Q, and dividing the surface of the sphere into new elements, beginning from Q as the origin, the element at P, with these new co-ordinates cosp and y, will be - dpdy. By reverting to the meaning of integration we see that the integral under consideration = (1-c) x limit of sum of all the elements of the surface of the sphere divided respectively by the cubes of their distances from D. = But this, by the change of co-ordinates, also 1 -1 2π (1 0 с (1-c) dpdy which = 2π (1 − c3) -1 1-c2 + const.) = 21-0 (1-10) C 1+ c =4π, whatever value c has. This coincides with the former result. 31. This integration helps us to see by what process c disappears from the result; and it will assist us in the latter part of the present demonstration. = The quantity 1-p is the versed-sine of the arc QP, and is measured along the line QCq. Let this line be divided into n parts each equal to h, so that n.h the diameter = 2, n being very large and h very small. Draw perpendiculars to the diameter through these divisions cutting the circle QPq in a series of points; and call the distances of these points from D, beginning from Q, 1-c, s', s", s'"...... s(n-1), 1+c. Suppose P is at the ath division; then and 1-p=x.h, d(1-p) or dp = (x + 1) h-xh=h. |