TRACT 27 . PLACE OF GREATEST ATTRACTION.

73

of

By Art. 4, the attraction

C * i ■, PF + PC ,

V PFCG IS 5 , PG X II. J. - ~ SX X I)

) PG

PGAH IS S

, , PH + pA n i ft

PG X h. 1.- — sx x ii. 1. —

PG

1 “g-te + W }

ax 3 f

. ]. —. C

a J

By Art. 5 , the attraction of efc is

EF . FC . PE

-- X (PC — PE —

EO ' 1 ^

EF , , FC ! +EC . l'C + l'E. tF SC

- X h.I.-=~ X

EC TE . EC + FF. . EF CS

[qq -g(a-.r) — gc x h.I.

a eg + eqq + aax— hex-

]•

g . (c+e) . (a— x) ■

Lastly, by Art. 6, the attraction of bde is

x (pb — PE

x h.

BE 2 + BE . BP*— PE . PEPE . BE—PE . DE

) *

(a — x) x (d — g + - x h. 1. - c + de

-eg.

eg

These quantities being collected together, with their pro-per signs, and contracted, we have ,

i X

ab . ad—qq-

-V C . -

’a ee

| cc g.{a-x) ^ j (cej

-itx - , . ag+qq — bz

-h x x hyp. log. — + j

de-

(i b+d) x- eg) . (otg + eqq + am — bex)

SS • (cc-cc) . (a-.*)

for the whole attraction in the direction fe.

9. Having now obtained a general formula for the mea-sure of the attraction, in any sort of triangle, if the parti-cular values of the letters be substituted, which any practi-cal case may require, and the fluxion of this attraction beput = 0, the root of the resulting equation will be the re-quired height, from the bottom of the hill.

10. But for a more particular solution in simpler terms,let us suppose the triangle abc to be isosceles, in which casewe shall have d=e, and g ~2l) ~ 2c, and then the above ge-neral formula will become

Qad—qq—dx

s X

+ t ?.2i>xh.l.

dd

X

&

b -f x x h. 1

2 ab + qq — bx

{b+d) . x2d6 a 4 dq^~~ (fe 2 —a 2 )26* . (a— x)

for the value of the attraction in the case of the isoscelestriangle, where f is = </ - A-ab'x + d'x-). And the

fluxion of this expression being equated to 0, the equation