RESULTING PROM THE THEORY OF THE GYROS COTE. 3 



-, being very great. In the phenomena of the gyroscope, the tirst condi- 



tion obtams ; in the case of the earth, attracted by the sun or moon (« being small), 

 it is easy to show that the alternative condition is fulfilled.' 



Putting - - equal to zero in equation (3) we get Q^a for the viaximnm of 0, and 

 ctt 



for the minimum, the equation, 



cos 6=—l3^-\-i/l-\-2l3^ cos «+/:(* 

 in which, if (3 is very great, the value of cos 6 differs but slightly from that of coso. 

 Hence by introducing a new variable a, equal to o — 0, and deducing the values of 

 dJ and (by development) of sin^Q and cos (neglecting the higher powers of v) and 

 substituting in (3) and (4), they become (omitting, as relatively small, cos w in the 

 factor cos cj-|-4.3^). 



5. . ^di^ 



■^ 2u sin a — 4o^tt* 



6. 'lj;=^3 



at 



' ^7i/ sniij 

 Equation (5) gives by integration and putting /? (^j^^^Jc, 



7. ?f^-— 5 sin ij sin "Jd 

 which substituted in (G) gives 



8. ^f=\h^mm 



dt /i* 



9. ^ 1 A-;__l sin2Z-i! 



2.i^ 4ii^ 



If we make o;=90°, sin cj=l, in equation (6), deduce the value of dt, and sub- 

 stitute in (5) we get, 



10. rl.],- ^^ 



I 1 



the differential equation of the cycloid, generated by a circle of which the diameter 



is -?i^o, and havin" a chordTi— «. 

 2,^^' =" 2ir 



' For the earth the moments of inertia, A and C, with reference to principal axes through tlie centre, 



71 \ C^ 

 (lilTer verv little. The value of 3 ma}' therefore be appro.\imatelv written ^ I , and the denominator 



is to be replaced (17) by Substitute the value of L (19) and put, for the sun, ' ^"1° (25), 



sin fl r' 



n I C 



and the value of i3 becomes — / , which is verv larffe, 



2<i,>/3(C-A)coso' ^ " 



The value as depending on the moon's attraction is (2Sa), — / L-TL'^! — of the same orde 



'^ ° ^ ^ 2n,>/3(C-^) cose 



of magnitude as before. 



