400 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



We must now consider the corrections which must be applied to this result; first, 

 for the effect which the resistance of the arm to motion has on the time of the 

 vibration : 2d, for the attraction of the weights on the arm : 3d, for their attraction 

 on the farther ball: 4th, for the attraction of the copper rods on the balls and arm: 

 5th, for the attraction of the case on the balls and arm: and 6th, for the alteration 

 of the attraction of the weights on the balls, according to the position of the acm, 

 and the effect which that has on the time of vibration. None of these corrections 

 indeed, except the last, are of much signification, but they ought not entirely to 

 be neglected. As to the first, it must be considered, that during the vibrations 

 of the arm and balls, part of the force is spent in accelerating the arm; and there- 

 fore, in order to find the force required to draw them out of their natural position, 

 we must find the proportion which the forces spent in accelerating the arm and 

 balls bear to each other. 



Let EDCec/c, fig. 4, be the arm; b and b the balls; cs the suspending wire. 

 The arm consists of 4 parts: first, a deal rod vcd, 73.3 inches long; 2d, the silver 

 wire Dcd, weighing 170 grains; 3d, the end pieces de and ed, to which the ivory 

 vernier is fastened, each of which weighs 45 grains; and 4th, some brass work cc, 

 at the centre. The deal rod, when dry, weighs 2320 grains, but when very damp, 

 as it commonly was during the experiments, weighs 2400; the transverse section is 

 of the shape represented in fig. 5 ; the thickness ba, and the dimensions of the 

 part DEed, being the same in all parts ; but the breadth b& diminishes gradually, 

 from the middle to the ends. The area of this section is .33 of a square inch at 

 the middle, and .146 at the end; therefore, if any points (fig. 4) be taken in 



cd, and 4 be called x, this rod weighs -— — '-—1 per inch at the middle; 



' cd ° 73.3 X .'238 r 



2400 x .14-6 ., , , 2400 w .33 - .184 x 3320 — 1848 * , , . 



73.3 x .238 at the end ' and TO X .238 ' = 73~3 at *'> and therefore, 



as the weight of the wire is — - per inch, the deal rod and wire together may be 



considered as a rod whose weight at x = rrr per inch. 



But the force required to accelerate any quantity of matter placed at x, is pro- 

 portional to a: 2 ; that is, it is to the force required to accelerate the same quantity 

 of matter placed at d, as a? 2 to 1 ; and therefore, if cd be called /, and x be sup- 

 posed to flow, the fluxion of the force required to accelerate the deal rod and wire 

 is proportional to ~ x ^^f 1848x) , the fluent of which, generated while x 



flows from c to d, is = — — X'-J ; • — -r- == 350; so tnat tne f° rce required to 



73.3 3 4 



accelerate each half of the deal rod and wire, is the same as is required to acce- 

 lerate 350 grains placed at d. 



The resistance to motion of each of the pieces de, is equal to that of 48 grains 

 placed at d\ as the distance of their centres of gravity from c is 38 inches. The 

 resistance of the brass work at the centre may be disregarded; and therefore the 



