120 AN EMPIRICAL STUDY OF GYRATING BODIES. 



Reversal. 1st inst. 2d inst. 3d inst. 4th inst. 5th inst. 



1st 0.766 : 0.133 : 0.023 : 0.003 : 0.000, etc. 



2d 0.766 : 0.133 : 0.023 : 0.003, etc. 



3d 0.766 : 0:133 : 0.023, etc. 



4th : 0.766 : 0.133, etc. 



5th : 0.766, etc. 



, etc. 



0.766 : 0.899 : 0.922 : 0.925 : 0.925, etc. 



If we had included the 6th instant, the result would 

 have been 0.925, and the same ever after. The effect in- 

 creases for a few terms by a constantly diminishing in- 

 crement, and then becomes constant. 



The formula vers °^ s s ? l2a gives the same final result in a 

 much shorter way, but does not exhibit so well the char- 

 acter of the series. 



I will now apply this formula to certain crucial 

 conditions : 



Suppose angle a=o. In other words, that the wheel 

 is reduced to an axle with weight, but no sensible 

 diameter : cos a=l ; cos 2a=l ; therefore the series be- 

 comes — the impulses being numbered as before : 





1st inst. 



2d inst. 



3d inst. 



4th inst. 



5th inst. 





1 



1 



1 



1 



1 



1 



etc. 



2 





1 



1 



1 



1 



etc. 



3 







1 



1 



1 



etc. 



4 









1 



1 



etc. 



5 











1 



etc. 



These series do not converge, and consequently they 

 give for each instant the old formula, V=tv, where 

 V'=velocity after the instants, and v=velocity at the end 

 of the first instant. 



Suppose angle a=45°. We have cos a=.71 ; cos 2a =0. 

 Therefore, the series become : 



0.71. 



























0.71. 

























0.71. 





 0.71 













0.71. 





 

 O&c. 



0.71; 



C.71; 



0.71; 



0.71 ; 



104 



0.71 



&c. 



