352 Astronomical ami Nautical Collections. 



of the velocity, might be sufficiently accurate for the purpose : 

 and Euler has inferred, from Newton's experiments, that the 

 constant resistance of the air to the motion of a leaden ball, 

 two inches in diameter, was about one millionth part of its 

 weight, or that it would cause it to remain at rest at an angu- 

 lar deviation of 0".2 from the vertical line : but a part at least 

 of this resistance may perhaps have been derived from the want 

 of flexibility or elasticity of the thread. 



From a mean of 60 experiments of Captain Kater, consisting 

 of about 5000 vibrations each, we obtain 1 °185, 1°.086, 0°.997, 

 0°.919, and 0°.843 for the successive values of the arcs, at 

 intervals of about 960 vibrations : and a slight irregularity in 

 the second differences of these numbers makes it probable that 

 .997 ought to be altered to .998. With this correction, the 

 successive diminutions, in about 1920 vibrations, will be .187, 

 ,167, and .154, for the respective arcs of intermediate values, 

 each of which must be supposed to exceed the intermediate arc 

 actually observed by one third of its deficiency below the mean 

 of the two neighbouring numbers, and we may call them 1.088, 

 1.000, and .9195, respectively. 



Putting then D = a* + A?/ + A-z, for the diminution of the 

 arc, we have three equations, the last of which, subtracted from 

 the first, gives us .1685 (y + 2.1075 z) ~ .033, and 

 y + 2.1075 z = .1958; consequently, if z zz 0, y = .196, 

 which would be the coefficient for a resistance simply propor- 

 tional to the arc, giving j; + .196 for the amount of the second 

 diminution, that is, .167 ; so that x would require to be negative, 

 which is impossible : and if 3/ = 0, z = .093, and the second 

 diminution would require x to be .074: a value which is suffi- 

 ciently compatible with these equations, but which would not 

 be applicable to the shorter vibrations ; an arc of 0.°80, for 

 example, exhibiting a diminution of about .11, and leaving only 

 about .050 for x, so that x must probably be still smaller than 

 .05, and if we make it = .040, we shall have .127 left for y+z, 

 and .196 - .127 — .069 = 1.1075 z, and z = .062, and 

 y = .065, and D = .040 + .065 A + .062 A\ which gives 

 .132 for an arc of .8, and :i- is still too large. Now, if we take 



