DYNAMICS. 



BD to H, making DU = AK, BH will be the apparent 

 motion of U. .loin All ; it is parallel to CD, Uv.m-e 

 HI) is = , and || AE. Hence the A AHB is similar 

 to the A CUB or ACF. In the sonic manner, the 

 -mall A, apparently described, and that actti.illx dc- 

 M-rilwd tin- next nioiurnt, will l>e similar; therefore the 

 i urves apparently described, nnd thuM- actually descri- 

 ked in any finite time, will be similar. 



PROP. XIX. 



The same supposition being made as in the two last 

 propositions, the times in which the bodies describe 

 their curves, will be to the time in which one of the 

 bodies would describe a similar curve around the other 

 restrained from moving, as the square root of the mass 

 of the other body to the square root of the sum of the 

 masses. 



Suppose that the similar arches described, BD, BH, 

 are inhnitely small ; let BG be a tangent, then FD 

 and GH will be the small deflections caused by the 

 same force in the two curves. Now, when a force 

 of pressure remains the same, the time is proportion- 

 al to the square root of the space. Hence v'FD : 

 i/GH : : time in FD : time in GH : : time in BD .- 

 time in BH ; but the figures being similar, v'FD : 

 VGH : : v'CB : V'AB; that is, : : -V/A : v'A+B. 

 Hence time in BD : time in BH : : v'A : v'A + B. 



The same ratio will hold with regard to the in xt two 

 small similar archc*, and hence with regard to any tl- 

 niu- similar arches. 



Con. Tin- Inxlies will describe areas proportional to 

 the times around the centre of gravity. For 1 . 

 proposition, the times of describing similar sector- 

 ing constant ratio ; and it is t-asy to see that .similar 

 sectors are in a constant ratio. But in the large figure, 

 the sector-, described are proportional to the time--. 

 (Prop. 1.) therefore this must hold also in the small fi- 

 gures. 



Supposing the body to have bulk, it can be shew u 

 that a force passing through its centre of inertia or of 

 gravity will produce the same effect as if the whole 

 mass were collected in that point, and that in any other 

 direction it will produce two motions, the one rotatory, 

 and the other progressive: See ROTATION. The discus- 

 sion of CENTRAL FORCES, given above, exhibits some 

 of the chief cases of the action of bodies on one ano- 

 ther at a distance. The article ASTRONOMY PHYSICAL 

 exhibits others. The action of bodies on one another in 

 contact will be given under IMPULSE ; and their ac- 

 tion on one another through the medium of a third, 

 will be seen under MKCII. \\ics. 



On Dynamics, the reader may consult La Grange's 

 Mechanimie Analylique ; D'Alembert'sDyiiamiyuc ; (ire. 

 gory's Mechanicf. (T. D.) 



Dynamo- 

 meter of 



onihaine 

 and Desa> 



gullets. 



PLATE 

 CCXUI. 

 Vig 1. 



D Y N 



DYNAMOMETER, from 3v 



a measure, is the name of an instrument ibr measuring 

 the relative strength of men and animals. 



The dynamometer invented by Mr George Grahame, 

 and improved by Dr Desaguliers, consists of a strong 

 frame of wood, A BCD. Through the piece DC is a 

 hole D, sufficiently large to admit a cylindrical iron 

 bar, about an inch in diameter. Upon this bar is a 

 square to receive the two separate and unequal arms of 

 a bent lever DF, DE, which are kept tight in their place 

 by a strong screw nut d. The arm DE, which carries 

 a weight W, is prevented from falling below a horizon- 

 tal position, by a metallic pin at K, which stops the arm 

 DF in its progress towards C, but both the arms move 

 freely in the opposite direction. At the top of the arm 

 DF, seen separately at dj\ is a round cross bar about 

 six inches long. The iron piece LN, also seen separately, 

 has likewise a cross bar at top, and holes for iron pins 

 to fasten it in its place. Another piece of iron HGI, 

 seen separately at h g i, is fastened to the timber that 

 carries the lever by a strong wooden screw at I, and by 

 the pin K going through its wings and the timber. 

 The collar S is to be put on when the upright arm of 

 the lever is not used, and M is the centre of gravity of 

 the steel-yard DE. 



In using this machine, the person who wishes to try 

 his strength, must take hold with his left hand of the 

 round part of the cross at N, and of the round part of 

 the cross at F with his right hand, and then by bring- 

 ing his right hand towards the left, in the direction 

 FN, he will move DE, and elevate the weight \V. 

 \Vhen this weight is lifted up so as to make the arm 

 ED just quit the pin at K, the force of the arm will be 

 determined in the following manner: Suppose the 

 weight W to be 56 pounds, and the distance from the 



D Y 



Dy nano- 

 meter. 



fulcrum, viz. WD, fifteen inches, then the momentum 

 of W will be 56 X 1 5 = 840. Let us suppose also, that 

 it requires six pounds applied at M, the centre of gravity 

 of the steelyard, to balance the steelyard itself. Then if 

 MD=10, we shall have (ix 10=(>0 for the additional 

 resistance made to the force of the arm ; so that the 

 whole resistance will be 8K>-|-6'0=900, which divided 

 by FDrrlO inches, the distance of the power, will give 

 ninety pounds for the force of the man's arms when ap- 

 plied at F and N. If another man is capable of raising 

 double the weight W, added to double the weight ot' 

 the steelyard at M, he will be twice as strong. In- 

 stead of increasing the weight at W, the weight may 

 be removed towards E. Desaguliers has described se- 

 veral variations in the construction of this machine, 

 and has also given a drawing and description of an in- 

 strument for measuring the strength of the fingers; but 

 for an account of these, we must refer the reader to 

 his Course of' EsjH-rimcntal Philosophy, vol. i. p. 29 1> 

 292. Annot. on Sect. iv. 



The dynamometer invented by Leroy of the Acade- 

 my of Sciences, consisted of a metal tube, ten or twelve namometerj 

 inches long, placed vertically on a stand, and contain- 

 ing a spiral spring, having above it a graduated shank 

 terminating in a globe. Tins shank, together with the 

 spring, was pressed into the tube in proportion to the 

 force which was applied to it, and pointed out upon the 

 graduated shank the strength of the person who ex- 

 erted the force. 



The most valuable dynamometer, however, is that negni' 

 which was invented by Regnier, and of which we have dynaroonw- 

 given a representation in Plate ( CXLI1. ir. 



This instrument, which resembles a common grapho- PI. AT J: 

 meter in its form and size, consists 01 a spiing AA rcxi.If. 

 twelve inches long, and bent into the form of an cilip- I * 



