i 



J. G, Barnard on the Gyroscope, 57 



That isj when tlie impressed velocity n, and in consequence i? 

 is very great, the minimnm value of 6 differs from its maximum 

 tt \)j an exceedingly minute quantity. 



Here then is the result, analytically found, which so surprises 

 the observer, and for which an explanation has been so much 

 sought and so variously given. The revolving bodj^, though 

 solicited by gravity, does not visibly fall. 



Knowing this fact, we may assume that the impressed velocity 

 n is very great, and hence cos 5— cos a exceedingly minute, and 

 on this su])position, obtain integrals of equations (6) and (7), 

 which will express with all requisite accuracy the true gyroscopic 

 motion. For this purpose, make 



6=1 a — u, dOzzz —du 



in which the new variable u is always extremely minute, and is 

 the angular descent of the axis of figure below its initial eleva- 

 tion. 



By developing and neglecting the powers of u superior to the 

 square, we have 



siii^ 6=is\n^ Of— 'wgin2a-j-?/2 cos 2a 

 cos ^ — cos a r= « sin cc — -^M 2 cos a 



substituting these values in eq. 6 we get 



du 

 V 2« sin a — m2 (cos «-|~'*i^^) ' 



^ having been assumed very great, cos « may be neglected in 

 comparison with 41^^^ and the above may be written 



du 



uaincc—i^'-^u 



{d) 



Integrating and observing that u = o^ when t — o^ we have 



* By Stirling's theorem, 



n u^ 



in which 17, U\ W &c. are the values of/ (m) and its different co-efficients ^hen u 



is made zero. 



Making /(w) = sin^ (a— u), and recollecting that sin2w=:2sin?ico3w and cos22/ = 



cos^M — sin^w, we get the value of sin^d; and making /(w)=co3(a— «)-C03a 

 the value in text of cos 5— cos a is obtained. 

 I Eq. 6 may be written 



% dO^ . (cos 9 — cos a)' 



-^ =2(00.6 -co,a)-i^' ^i^^ 



By substituting the values just found, of dd, sin^ ^ and cos 5— cos a and per- 

 forming the operations indicated, neglecting the higher powers of w, (by which 



(cos $ — cos a) 



3 



sin^^ 



reduces simply to i*') and deducing the value I £ di, the exprea- 



Bion in the text, is obtained. 



SECOND SERIES, VOL. XXIV, NO. 70.— JULY, 1857. 



8 



