A Treatise On Infinitesimal Calculus: Containing Differential And ..., volume 4

Bartholomew Price

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A(8in<^^ + co9^) + (c-A)n(«a8in<^-a»,co80) = ; (152) but from (104), Art. 42, dylf a>i sin <^ + a>2 cos <^ = sm^-^; therefore 8m^-^+ co8^-^ = -^ (sin 0-j^)-Kco8*-«,sin.^)-^ by reason of (102), Art. 42 ; and substituting these in (152), we and multiplying by sin Oj and integrating, we have A (sinj^)* ^ + en cos^ = A'; (154) where h' is a constant introduced in integration, and is indeed the sum of twice the product of each particle and the projection on the horizontal plane of (a?, y) of the sectorial area described by its radius vector in an unit of time. For although the prin- ciple of conservation of moments or of areas, see Art. 58, is not true of this motion relatively to any plane, yet it is true for the plane of (a?, y), because the axis of z is parallel to the line of gravity, and the weight consequently does not produce any moment relatively to that axis. Thus, of equations (70) in Art. 58, the third is true, and (154) is the form of its integral in this case. For A^idtj At^^dty en dt sre respectively the sum of twice the products of each particle and the sectorial area de- scribed by its radius vector in dt in the planes of (rj, 0> (£ f )> (f , »?) respectively ; and the areas described on the plane of (x, y) are the sums of the projections of these on that plane ; and therefore = {Aa>iSin^sin4> + Aa>2sindcos<^ + cncos^} dt and this is constant by the principle of conservation of areas ; so that if V is the sum of twice the product of each particle, and the protection on the horizontal plane of the sectorial area, described by its radius vector in an unit of time, (154) is the particular form assumed by the principle in this problem.

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