Vibrations of Heavy Bodies. 99 



as the only retarding cause. This must, according to every 

 theoretical principle, be taken to vary as the squares of the 



velocities. Then in passing through any small space z, the di- 

 minution of velocity (^) will be proportionate to v^ the square 

 of the velocity, and to t the time, but 

 < — _f. .". <p = v^ x_f. = i)z or to the space multiplied by 



V V 



the velocity of movement through it. 



Now, as before in fig. 2d, the velocity at w will be "* ~ ^^ 

 And consequently this multiplied by — 2x will be = — ^ 



By expanding aa — x^^ and changing the signs 



^=:J2xax(l-— — - — — - — — &c.)xiand 

 ^ 2a? 8 a* 48 a6 ^ 



^ ^ 2 3 a2 8 5 a* 48 



-i^&c.) 

 7 a6 ^ 



When a; = a (p should be equal to nothing, but the equation 



then becomes C + J 2 . (a^ — — a* — — a" &c.) 



^^23 85 



therefore C = - ^2 (a» - — . — a^ - JL . J- a=&c.) 



2 3 8 5 



When X = the variable terms vanish, and the equation be- 

 comes ffl = — */ 2 (a* — — a- — — . — a" &c.) 



^ ^^23 85 



The diminution of velocity is therefore proportionate to the 

 square of the arc. And if w = the velocity due to any arc of 

 descent a, the actual velocity, when it is performed, will be 

 V — a"^ V. The ascent due to this velocity will be v- — 2a- w? 

 but the arcs being as the square root of the ascent, the arc due 

 to the velocity will be v — a- v ; therefore the diminutions of 

 the arcs are proportionate to the squares of their length. 



Vol. XV. H 



