606 PHILOSOPHICAL TRANSACTIONS. ' [aNNO 1780'. 



11. To find the maximum of the expression in the last article, put its fluxion 

 = O, and there will result this equation 



' + V(T^M^) -^'^•^- -j;, -^ ''- '• T—n ; the root 



of which is n = -251999. Which shows that, in the equilateral triangle, the 

 height from the bottom to the point of greatest attraction, is only -5^-5- part more 

 than i of the whole altitude of the triangle. And this is the limit for the 

 steepest kind of hills. 



1 2. Let us find now the particular values of the measure of attraction arising 

 by taking certain values of ?i varying by some small difference, in order to dis- 

 cover what part of the greatest attraction is wanting by observing at different 

 altitudes. And first using the value of n, viz. •251999, ^s found in the last 

 article, the general formula in art. 10, gives ^a X 1 '07 (33700, for the measure 

 of the greatest attraction. — Again, if ?i = ^'-^, or :r = -^a; the same formula 



gives f^^ X :(17 -/79 + 6h.l.l^^-5 + ^h.l.ii±l^I^)=.«Xi-07025l2, 

 for the attraction at -^o( the altitude; which is something less than the other.— 

 And if w =: -jig- =: ^ ; the formula gives 



g X {16 - k/ 76 + 8h.l ^±^ + 3hA.l±^^) = sa X 1-0224232, for the 

 attraction at -jSj- or f of the altitude; less again than the last was. — Also, if w := 

 -jij. = -1-; the formula gives J-sa X [3 — -/S — 2 h. 1. 3 + -j-h. 1. (3 + 2^/3)} 

 = 5a X -9340963, for the attraction at halfway up the hill; still less again than 

 the last. — Further, if w = -^ = ^; the formula gives 



gx (14-^/76+ l2h.l'^-^^ + 2h.l^-±f^)=saX -8109843, for the 

 attraction at ~\ or f of the altitude from the bottom ; being still less than the last 

 was. And thus the quantity of attraction is continually less and less the higher 

 we ascend up the hill above the -251999 part, or in round numbers .252 part of 

 the altitude. Let us now descend, by trying the numbers below -252. 



And 1st, if w = -25 = i^; the same formula in art. 10 gives i^a X (7 — \/13 

 + 2 h. 1. l±^^ + -i-h. 1. i±i£l^) =sax 1-0763589, for the attraction at ^ 

 of the altitude; and is very little less than the maximum. — If n = -^Sj. = -l; the 

 formula gives -J^sa X (9—^/21 + 2 h. 1. 



9.±l^+2h.l.^-±|^)=.,VaX(9-^21+2h.l.^-^<ii) =sa X 

 1-0684622, for the attraction at -^\ or -f of the altitude; and is something less 

 than at J- of the altitude. — If n = -5^^; the formula gives 



5 X (l9-v/91+^h.l.l^^ + ih.l.^i±|:^) =.a X -9986188, for 



the attraction at -pV of the altitude; still less than the last was. — And, lastly, if 

 n = O, or the point be at the bottom of the hill; the formula gives -^sa x (2 + 



