Page:Scientific Memoirs, Vol. 2 (1841).djvu/217

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If we combine, then, with these propositions, the known theorem, that every function of $$\lambda$$ and $$u$$, which, for all values of $$\lambda$$, from 0 to 360°, and of $$u$$, from to 180°, has a determinate finite value, may be developed into a series of the form the general member of which, $$P^{n,}$$ satisfies the above partial differential equation,—that such a developement is only possible in one determinate manner,—and that this series always converges,—we obtain the following remarkable propositions.

I. The knowledge of the value of $$V$$ at all points of the earth's surface is sufficient to enable us to deduce the general expression of $$V$$ for all external space, and thus to determine the forces $$X$$, $$Y$$, $$Z$$, not only on the surface of the earth, but also for all external space.

It is clearly only necessary for this purpose to develope $$\frac$$into a series according to the above-mentioned theorem.

In the sequel, therefore, unless it is expressly stated otherwise, the symbol $$V$$ is always to be taken as limited to the surface of the earth, or as that function of $$\lambda$$, and $$u$$ which follows from the general expression, when $$r$$ is made $$= R$$: thus

II. The knowledge of the value of $$X$$ at all points of the earth's surface is sufficient to obtain all that has been referred to in Prop. I. In fact, according to Art. 15, the integral $$\int_0^u X\,d\,u = \frac$$ signifying the value of $$V$$ at the north pole, and the developement of $$\int_0^uX\,d\,u$$ into a series of the form referred to must necessarily be identical with

III. In like manner, and under the considerations in Art. 16, it is clear that the knowledge of $$Y$$ on the whole earth, combined with the knowledge of $$X$$ at all points of a line Rh