414 I'HILOSOl'HICAL TRANSACTIONS. [aNNO 1778. 



• OH 



instead of the quantity — X c in the theorem above, it will become be X c^ for 

 the measure of the attraction of the pillar whose base is bd in the direction an. 

 Which is as easy and simple an expression for the attraction of a single pillar as 

 can well be desired or expected. 



But to make the application of this theorem still n)ore easy to the great number 

 of small pillars concerned in this business, let us suppose be and d to be constant 

 or invariable quantities, and then it is evident that we shall have nothing more 

 to do but to collect all the i's or sines of elevation of all the pillars into one sum, 

 and then multiply that sum by the constant quantity be X d, by which there 

 will be produced the measure of the attraction of all the pillars, or of the whole 

 part of the ground on one side of we. Now be will be made to become con- 

 stant, by making the circles equi-distant from each other, or by taking the radii 

 in arithmetical progression. And d will be constant, by drawing the radii so as 

 to form with an angles whose sines shall be in arithmetical progression; for then 

 d is the common ditFerence of the sines of those angles. Hence then we are 

 easily led to the best manner of dividing the plan into the small spaces, viz. from 

 the centre a describe a sufficient number of concentric and equidistant circles ; 

 divide the radius ai of any one of them into a sufficient number of equal parts, 

 and from the points of division erect perpendiculars to meet the circle ; then 

 through the points of intersection draw radii, and they will divide the circles in 

 the manner required. 



In a computation of this kind, we need only calculate the attraction of the 

 matter above the plane or horizon of each observatory, and the attraction of so 

 much matter as is wanting to fill up the vacuity below that plane lying between 

 it and the surface of the lower part of the hill. For the south observatory, the 

 attraction of the southern parts that are above it must be subtracted from that of 

 the northern parts, to obtain the attraction of the whole towards the north ; that 

 is, the southern elevations are negative, and the northern ones affirmative 

 The contrary names take place with respect to the depressions, or the vacuities 

 below the plane of the observatory ; for if the whole space below this horizontal 

 plane were full of matter to an equal extent both ways, its attraction need not be 

 computed, as those on the contrary sides would mutually balance each other ; 

 but since there are unequal vacuities on each side, it is evident, that the attrac- 

 tion of the matter that might be contained in them must be deducted from the 

 other two equal quantities, to leave the real attraction of those two sides ; then 

 subtracting the remainder on the south side from that of the northern side, 

 there will at last remain the joint effect of all the matter below the plane in the 

 northern direction : but as the one remainder is to be subtracted from the other, 

 the two equal quantities may be omitted in both, and only the effects of the va- 

 cuities brought into the account, which being twice subtractcil, their signs be- 



