12

DETERMINATION OF THE

TRACT 2T.

mathematically true, and yet at the same time sufficientlyexact for the before-said limited extent also. It will alsocome nearest to the practical experiment, to suppose thehill to be a long triangular prism, so that all its meridionalsections may be similar triangles. Let therefore the triangleABC represent its section,by a vertical plane pass-ing through the meridian,or one side of an indefi-nitely thin cuneus, whoseedge is in pg ; or ratherFBCGP the side of one cu-neus, and pag the side ofanother, their common edge being the line pg, perpendi-cular to the base Ac; p being the required point in the sideAB, where the attraction of the section ABC, or indefinitelythin cuneus, shall be greatest, in a direction parallel to thehorizon Ac. And then, from the foregoing suppositions, itis evident that, in whatever point of ab the attraction of abcis greatest, there also will the attraction of the whole hill bealso the greatest, very nearly.

8. Now r draw hpdef parallel to the base Ac; and ah,pg, bi, cf, perpendicular to the same. Then it is evidentthat at the point p, in the direction pf, the attraction ofpbcgp is affirmative, and that of pag negative. But pbcgpis = FED + BDE + PFCG — EFC ; and PAG = PHAG — PHA.Therefore the attractions of pbd, bde, Pfcg, pha, are af-firmative ; and those of efc, phag, negative.

Put now bi — fl, Ai — b, ic ~c, ab = d, sere, AC — g— b-\-c, and pg = x, the altitude of the point p above thebottom. Also let s — the sine of the indefinitely small angleof the cuneus, to radius 1; and q*=</(a 1 g* —2abgx-\-d z x' 1 ).

Then by Art. 3, the attraction

H_P.

of

... BD 7 — X

PBD IS 5 . PD . — = Sb X — t,bp a *

PHA is S . PH . — — sb X 4-.

FA a

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