DETERMINATION OF THE

74

tract 2t,

will give the relation between a and x, for any values ofb and d, by a process not very troublesome.

11 . Now it is probable that the relation between a andwhen the attraction is greatest, will vary with the variousrelations between b and d, or between b and a. Let ustherefore find the limits of that relation, between which itmay always be taken, by using two particular extremecases, the one in which the hill is very steep, and the otherin which it is very flat, or a very small in respect of b or d.

12. And first let us suppose the triangular section to beequilateral; in which case the angle of elevation is 60°,which being a degree of steepness that can scarcely everhappen, this may be accounted the first extreme case.Here then v r e shall have d — 2b — -§- a*/3 , and the formula

in Art. 10, will become $ x (——- + x X h. 1. —- +

o—x , i <i + 2r + r.

- 7 - x h. J» —

4 a

), for the value of the attraction in thecase of the equilateral triangle, in which r is = — ax

+

13. Or if we take x 3 na, where n expresses what, partof a is denoted by x, the last formula will become sa X

Ci -» + »■)+.. x h. + 1=

x h. 1. 1 + ”+-v'h—^ } f or the case of the equilateral tri-

angle.

14. To find the maximum of the expression in the lastarticle, put its fluxion = 0 , and there will result this equa-

tion > 1 + ^cr^= 2 h yp- Io s

2 — J1 + 2 yO-ri + ,i’)3,a

- L h. 1.

1 +»! + 2^/0 — n + 11 2 ]

; the root of which is n = . 251999 . Which

shows that, in the equilateral triangle, the height from thebottom, to the point pf greatest attraction, is only - 5 - 5 -othpart more than q of the whole altitude of the triangle.And this is the limit for the steepest hind of hills.

15. Let us find now the particular values of the measureof attraction, arising by taking certain values of n, varying