On the grounds of the method which Laplace has given in ... his 'Mécanique céste' for computing the attractions of spheroids of every description / [Sir James Ivory].

Ivory, James, Sir, 1765-1842.

Date:: 1812

Credit: On the grounds of the method which Laplace has given in ... his 'Mécanique céste' for computing the attractions of spheroids of every description / [Sir James Ivory]. Source: Wellcome Collection.

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$first position, will be = p >/2(1—7); and f, the same dis¬ tance in the second position, will be = j (p.-J- ^r)2— 2p(p + <5/') • y + p2}Y; snd if, with Laplace, we neglect the square and other higher powers of <5V, then f'~ [ 1 \ • —} •/: there¬ of f dm ~P j'dm — consequently*'—— P 1/ Pd?n -J-7 f h 2? fore x/ Since the spheroid is supposed to approach very nearly to the spherical figure, the radius of it will fall under this form of expression, viz. p = ax(i-{-a.y); where a denotes the radius of a sphere concentric with the spheroid and nearly equal to it; a, a coefficient so small that its square and other higher powers may be neglected ; and y, a function of two angles 6 and which determine the position of p, 6 being the angle contained by p and a fixt axis passing through the centre of the spheroid, and -w the angle which the plane drawn through p and the axis, makes with another plane passing by the same axis. Now, by substituting and neglecting all the terms of the order u and the higher orders, the preceding values of V and dV_ dr will become V = ?•<■*. (1 + 2«.y) +fj dV dr dm 7 and, by combining these so as to exterminate^we shall get *v+«(£) 2 w „ — . a 3 which is no other than Laplace’s equation.* •e.$