Page:Mécanique céleste Vol 1.djvu/35

I. i. §1.] $$z'$$ differs from $$z$$ but by an infinitely small quantity; again, $$dx''$$ forming with $$dx$$, the angle $${\pi\over 2} - \theta$$, we shall have

therefore

^ —ydx -. -^^1"^

d6=-^-^-. ., r.„„ r.., ,-. [7]

If we vary the force $$y$$ by $$dy$$, supposing $$x$$ constant, we shall have the corresponding variation of the angle $$\theta$$, by changing in the preceding equation $$x$$ into $$y$$, $$y$$ into $$x$$, $$\theta$$ into $${\pi\over 2} - \theta$$, which gives

J. xdy

supposing, therefore, $$x$$ and $$y$$ to vary at the same time, the whole variation of the angle $$\theta$$ will be — ^ — | — ; and we shall have

Substituting for z^ its value 3^ -{-if, and integrating, we shall have

to the powers of d6, by Taylor's Theorem [617], or by any other way, will be of the form A — k.dd-{-J(/ .d(P — etc. — A;, /fc', etc. being constant quantities, dependant on the first, second, etc. differentials of (p {^nr). By this means, do/' [4] will become d3cf' = z' .{A — kdd-{-^d(P — etc.]. Now it is evident, that when dx = 0, the quantities da/' and dd must also vanish; and the preceding expression will, in this case, become = z .A, or ^ = 0. Substituting this value of A, we get generally djc"= 2/^ — k d 6 -- Jcf d (f^ — etc.} ; and by neglecting the second and higher powers of d &j it becomes as above, dx" = — kdd .z'.


 * (8) As in note 5.

f (9) By putting the two values of $$dx''$$ [5, 6,] equal to each other, and deducing therefrom the value of $$d\theta$$.

J (10) As d is changed into - — 6, the differential dd changes into — dd.

§ (I J ) By die substitution of ar^ + ^ for z^, the equation becomes — ^—7^ — = kd6'j

xx--yy

"{^

or, as It may be written, -— - =:kdL for the differential of the numerator of the first

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