VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. t5oy 



h. 1. 3) = sa X 7746531, for the attraction at the bottom of the hill; which is 

 between f and f of the greatest attraction, being sometliing greater than ^ but 

 less than 4 of it. 



13. The annexed table exhibits a summary of the 

 calculations made in the preceding cases; where the 1st 

 column shows at what part of the altitude of the hill 

 the observation is made; the 2d column contains the 

 corresponding numbers which are proportional to the 

 attraction; and the 3d column shows what part of the 



greatest attraction is lost at each respective place of observation, or how much 

 each is less than the greatest. 



14. Having now so fully illustrated the case of the first extreme, or limit, let 

 us search what is the limit for the other extreme, that is, when the hill is very 

 low or flat. In this case b is nearly equal to d, and they are both very great in 

 respect of a; consequently the formula for the attraction in art. 9, will become 



barely s X {x X h.l. — y 1 {a — x) X h. 1. ^^-—) ; the fluxion of which 



being put = O, we obtain O = h. 1. 



2 h. 1. = h. 1. h. 1. ( )- = h. 1. ->-- — '—; hence therefore 



X a—x X ^a — x' x(2a — .i) 



(a — xf =.x {2a — x), and ar=aX(l— V-i-) = •2929a. Which shows that 

 the other limit is -^„V; that is, when the hill is extremely low, the point of 

 greatest attraction is at -pVo of the altitude, like as it is at -jVs- when the hill is 

 very steep. And between these limits it is always found, it being nearer to the 

 one or the other of them, as the hill is flatter or steeper. 



15. Thus then we find, that at J- of the altitude, or very little more, is the 

 best place for observation, to have the greatest attraction from a hill in the form 

 of a triangular prism of an indefinite length. But when its length is limited, 

 the point of greatest attraction will descend a little lower, and the shorter the 

 hill is, the lower will that point descend. For the same reason, all pyramidal 

 hills have their place of greatest attraction a little below that above determined. 

 But if the hill have a considerable space flat at the top, after the manner of a 

 frustum, then the said point will be a little higher than as above found. Com- 

 monly, however, J- of the altitude may be used for the best place of observation, 

 as the point of greatest attraction will seldom differ sensibly from that place. 

 And when uncommon circumstances may produce a difference too great to be 

 entirely neglected, the observer must exercise his judgment in guessing at the 

 necessary change he ought to make in the place of observation, so as to obtain 

 the best effect which the concomitant circumstances will admit of. 



