



C. 



B. WARRING. 





119 



pulses. 



1st inst. 



2d inst. 



3d inst. 4th inst. 



. r itli inst. 





1st. 



cos a : 



cos a cos 2a 



: cos a cos 2 2a : cos a cos ;i 2a 



cos a cos 4 2a 



etc. 



2d 





cos a 



: cos a cos 2a : cos a cos 2 2a 



cos a <:• 



etc. 



3d 







cos a : cos a cos 2a 



cos a cos 2 2a 



etc. 



4th 







cos a 



cos a cos 2a 



etc. 



5th 









cos a 



: etc. 



The terms for any one instant are the same as the 

 terms of the first series up to that point. 



The combined effect at the end of the fifth instant 

 equals the sum of five terms of the series. When a is 

 large, and, consequently, cos 2a very small, the series 

 converges rapidly. By the usual method of obtaining 

 the sum of a descending geometrical series, we find the 

 ultimate velocity to be equal to ,-dlrk, or Tere ^ n3a . 



This last is a convenient formula for those whose tables 

 give the versed sines, but for others, it should be changed 



j-„ cos a 

 LU 2 sin 2 a- 



But this, it will be remembered, is to be multiplied by 

 Am, fig. 8, the velocity imparted by gravity to A during 

 the time, t, of one-quarter of a revolution. Calling this 

 velocity d, we have for the falling rate, q, of a gyroscope 

 in action, q = ve ? sed °ln 2a 5 or > since d varies as the time it is 

 falling, and that varies inversely as the angular velocity 

 of the wheel w, we have q = verse d° S in2a,w ; or > if w © make 

 n = number of revolutions per second, we can write our 

 formula, q = n ver c s ° 3 d a S in2a : Hence, q varies inversely as n ; 

 and, if n is infinite, q = = o = a result which agrees with 

 a previous one, and with experiment, so far as experi- 

 ment is possible. If n is uniform, then qis uniform. 



This is in accord with the result when discussing the 

 case in which the angle was only 45°. We have now 

 reached the same conclusion when a equals any angle. 



It will be not without advantage to work out numeri- 

 cally a single example. We will take the most com- 

 mon of all cases — that in which the angle a is less than 

 45°— let it be 40°. 



Our series gives for the impulses : 



103 



