C. B. WARRING. 121 



In other words, there is no increment whatever. 



Total for any instant, = verS ed 3 s fn a a = verse°dsin9o = _ r~ = 0.71 ; 



If a = 90°. 



Then we have ^S = 0. 



Let it be any angle between 45° and 90° ; then, since, 

 in our series, every other term is negative, it may be 

 thought that, possibly, the sum of the negative terms 

 may exceed the sum of the positive, and hence, that the 

 gyroscope ought sometimes to rise. 



The total effect at any instant has been shown to be 

 equal to ^"n^ . As a is less than 90°, cos a must al- 

 ways be positive, and the versed sin is always positive ; 

 hence the value of the fraction is always positive, and 

 in no case can there be an upward movement, as the re- 

 sult of the reversals 1 . 



There is a case where the gyroscope actually rises to 

 a vertical, but it depends on another principle and other 

 conditions, which will be considered hereafter. 



I have, thus far, considered only the case in which the 

 axis of the gyroscope is horizontal. If the inclination is 

 small no sensible error can arise from such neglect. 



But suppose the inclination to be any angle b ; I find 

 by a process similar to that already employed that the 

 total effect is ^fjfits I olir previous formula multiplied 

 by the cosine of the inclination. The effect, therefore, 

 of the axis being above or below the horizontal is to 

 reduce the falling rate, since cos b is always less than 

 unity, save when 5 = 0°. 



It follows that there is from this cause an increase of 

 rate of gyration as the axis approaches the horizontal, 

 and a decrease as it goes from it. 2 



We will now apply our results to the actual gyro- 

 scope, to see how its fall corresponds with our theory. 



IThe oscillations sometimes seen when a gyroscope is started, especially by unskillful 

 hands, are accidental, and have no influence in sustaining it. 



2 The clearest and best experimental demonstration is afforded by the top in that series 

 of cases, hereafter considered, in which the inclination of the axis is alternately increased 

 and decreased. 105 



