382 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The distance J] the actual length of the fall, as measured by a steel 

 tape which was tested by a Brown and Sharpe steel meter rod, was 

 2285 cm. Accordingly S = 36 cm., and the easterly deviation should 

 be, according to Gauss, 



6 28 

 y = % cos 42° 22' X — '- — X 2.176 (2285 — 18) = 0.177 cm., 



J 3 86400 v 



that is, to the third place of decimals the value of the easterly deviation 

 is not in our case affected by the resistance of the air, if I have cor- 

 rectly understood and used the formulas of Gauss. 



Coefficient of Air Resistance. 



It is perhaps worth while, since observations on the air resistance 

 offered to the motion of spherical bodies are not over numerous, to 

 work out from the data here at hand the coefficient of this resistance 

 for the spheres here used, — bronze spheres, one inch in diameter, 

 ground to a smooth surface, but left in a slightly greasy condition by 

 their experience of being dropped into beds of tallow in their use six 

 years ago. 



The mere buoyant effect of air on bronze may properly be neglected 

 in this discussion, as it is very small. 



If we assume that the resistance of the air is proportional to the 

 square of the velocity of the falling sphere, within the moderate range 

 of velocity here considered, we have, as the net accelerating force on a 

 ball of m grams, (mg — At 2 ) dynes, where k is the constant coefficient 

 of resistance. Accordingly, writing c for m -J- k, we find as the incre- 

 ment of velocity 



dv 



whence 



fe== (*-£)*' (1) 



dr cdv . . , 



= dt. (2) 



/•- (qc — v 2 ) 

 9~~ 



This equation, integrated for v between the limits and v, and for 

 t between the limits and 2.176 (the observed value), gives 



• r log ^±i:T =i |/5 log ^fe =2 . 17 6. ( 3) 



2 Vgc L ' Vgc -'-Jo 9 Vgc — v 



