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
Catalogue details

Licence: Public Domain Mark

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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    viz. ppyr—f-”1 . p1. v'. dp' . dm' |r*-2rp.7+P4]l±i when it is extended to the whole surface of the sphere, and in the particular circumstance of r == p, or r — p = o. We must begin with transforming the formula to be integrated. The arcs 9 and 9’ are the two sides of a triangle formed on the surface of a sphere; the angle contained by those sides is ■ar'— tct; and the third side of the same triangle is no other than the arc whose cosine has been denoted by y: let <p de¬ note the angle opposite to the side 9' whose cosine is p; then if we suppose 9' and w' to vary, it has already been proved that the correspondent fluxion of the surface of the sphere will be = pe. dp . d'sr'; but if we make y and q> vary, the same fluxion will be = pa. dy . dq>: therefore n (r—p)* 1 . p2 . v . dp . dm' _ (r—p)1 1 . p2. v’. dy . dtp # | ra—2/p . | r2—2rp . and as this is true for every element of the spherical surface, the fluents will likewise be equal when they are extended to the whole surface of the sphere. To complete the transforma¬ tion we must next convert t/ into a function of y and <p ; after which the integration with regard to q> will be independent of the denominator in which y only is contained. Suppose v' to be actually transformed as here mentioned, then PP(r—f)l~l . p2 . dp' . dm' p[r—f)? 1 . pa -dy .f v . d<p . ^ jr*-zrp.y+|r2_zrp .y+pa the sign of integration in the numerator being understood to affect the variable <p only.
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