VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 605 



tion of PBCGP is affirmative, and that of pag negative. But pbcgp is = pbd + 

 BDE + PFCG — EFC; and PAG = PHAG — PHA. Therefore the attractions of 

 PBD, BDE, PFCG, PHA, are affirmative; and those of efc, phag, negative. Put- 

 ting now BI = O, Al = Z), IC =: C, AB = </, BC =: e, AC =z g =^ b -\- C, and PG 



= X, the altitude of the point p above the bottom : also s ^ the sine of the inr 



definitely small angle of the cuneus to rad. I ; and q'^ =. >/ {a'g- — labgx -f (Px"^). 



Then by the foregoing articles, these several quantities being collected together 



with their proper signs, and contracted, there is at length obtained the expression, 



c V f^-l-c '"'-19-d' : I ^ V h ] <'I±ll-± + "S(''-^ } X h 1 

 *X L7 + '^- ee h^xn.i. ^^_^^^^ -t- ^ xn. 1. 



(ee+de-cg).(a,g+eqg + a^s-bcx)^ ^^^ ^^^ ^j^^,^ attraction in the direction pe. 



gg(ee—cc)x{a—x) -■ 



Q. Having now obtained a general formula for the measure of the attraction 

 in any sort of triangle, if the particular values of the letters be substituted which 

 any practical case may require, and the fluxion of this attraction be put = O, 

 the root of the resulting equation will be the required height from the bottom 

 of the hill. But for a more particular solution in simpler terms, let us suppose 

 the triangle abc to be isosceles, in which case we shall have d = e, and g =. 2b 

 ^ 2c, and then the above general formula will become 



^X I dd b-i-xxn.l. ^^_^^^^ -t ^^ .zu xn.i. 2A^(«-j) J' 



for the value of the attraction in the case of the isosceles triangle, where q'^ is 

 = ^{4a-b^ — Aab'^x + drx-). And the fluxion of this expression being equated 

 to O, the equation will give the relation between a and x for any values of b and 

 d, by a process not very troublesome. 



10. Now it is probable that the relation between a and x, when the attraction 

 is greatest, will vary with the various relations between b and d, or between b 

 and a. Let us therefore find the limits of that relation, between which it may 

 always be taken, by using 1 particular extreme cases, the one in which the hill 

 is very steep, and the other in which it is very flat, or a very small in respect of 

 b or d. And first let us suppose the triangular section to be equilateral; in which 

 case the angle of elevation is 6o°; which being a degree of steepness that can 

 scarcely ever happen, this may be accounted the first extreme case. Here then 

 we shall have d = 26 = fa \/ 3, and the formula in art. g, will become s X 

 2a-r_x _,. ^ >^ h.l. ^.^IILZI + ^Zf X h.l.^±^^") for the value of the at- 

 traction in the case of the equilateral triangle, in which r is = \/ («^ — ax -\- x'^). 



Or if we take x = na, where 7i expresses what part of a is denoted by x, 

 the last formula will become sa X (l — \n— \y/{\ — n + ri') + n X h. 1. 

 2-« + 2V(l-n + «^) _^ Lzf X h.l. l+" + ^Wl-" + «-K f^^ ^^g ^^^^ ^f ^^^ i,^, 



3n ^4 l-« ' ^ 



teral triangle. 



