Mr Forbes's description of a New Anemometer. 35 



relative density of air ; we have r in pounds = — -^ X -^--7- 

 •^ ^ 840 144 X4j 



ad^ ad^ v^ v^ d^ 



^^ = 4928| ^^'- ^"^ s^ibstituting --- for «, we have -^j^^-^. 



Whence to obtain a unit of resistance, putting v and d equal 



to 1 respectively, we have R = ^r^xriy ^^^ ^^ ^^^^ ^J 



/ W . 

 the equation already found u = ^ ^r";^' ^^^^^ ^ varies with 



the square of the diameter of the ball. Hence also 9> a =:i — y- 



Let us now endeavour to apply the principles thus obtained 

 to the measurement of velocities and spaces in a medium resisting 

 as the square of the former, since we have already shown, that, 

 by finding the effect of resistance in retarding independent mo- 

 tions, we shall have no difficulty in applying it to the simple 

 effect of resistance upon a body at rest. Since the velocity thus 

 resisted must be in a state of perpetual flux, we must find 

 its relation to the retarding force by some theorem of variable 

 motion. It is this ; ± d v —fd t. Now, since the resist- 

 ing force / is measured by the differential of the velocity r;, 

 that differential must be proportional to v"-^. Now this is the 

 characteristic of lines drawn from the assymptotes of a rectangu- 

 lar hyperbola to the curve, such as E F, G H, A B, Fig. 1. 

 Plate II. For let the abscissae D B, D H, &c. be denoted by 

 X, and the corresponding ordinates by «/, then, from the pro- 

 perty of equality in the rectangular spaces, we have the equa- 

 tion X y -zz c^ \ a being a constant quantity. Hence the diffe- 

 rential Ys, X dy -^ y d X — ^ and d y •=. — - — . But since 



— — — idyz=i — — —— in which dx beinej a constant differ- 



ential, (for, as we shall presently see, the abscissas increase uni- 

 formly,) and a^ a constant quantity, d^ must vary as ?/^, which 

 is property of the velocities resisted as their squares.* 



Hence, therefore, if we draw a line A B, which shall repre- 

 sent the initial velocity V, and one at right angles, B D cor- 

 responding to the time e, or the time of extinction of the velo- 

 city V, when uniformly resisted, the parallelogram thus formed 



* See note to Princip. ii. 5. ; de Motu corporum quibus resistitur in 

 duplicata ratione velocitatum. — Jesuit's Edit. 



