58 /- G. Barnard on the Gyroscope, 



, 1 r - 4pi^.* 



< = — - . arc f cos r=:l ^ — I * 



2^ } sin 



sin a , , ^ 



N 



or, (since cos 2a — 1—2 sin^ a) 



1 17 



« = ^sinBsiii2i? |^-.< (9.) 



Putting a — tt in place of Q (equat. 7) neglecting sq[uare of w, we get 



dxp 1 1^ 



dt /?^A 



.sin2(?Jf.f (10)t 



from wHcli, observing tliat V^ = 0, wten t = 



These three expressions (9), (10), (11), represent the vertical 

 angular depression — the horizontal angular velocity — and the 



2(3 4p2 



» 



/ - • may be put in the form — — - i 



V2w sin a -4)3 2^2 ^ ^ sm a 



du 



slDa 



2wt-^— m2 



8in« 



Call -7-7 =11, and the integral of the 2d factor of the above is the arc whose radiu8 

 is R and versed sine is w; or whose cosine h'R — u; or it is II times the arc whose 

 cosine 1 — k "with radius unity. Substituting the value of R in the integral and 



multiplying by the factor -: — we get the value of |~^ t, of the text. 



f In eq. (7) if we divide both members by sin^ 6, and, in reducing the fraction 



cos 9 - C03 a 



-r-g fl — » use the values already found and neglect the square, as well as higher 



powers M, (which may be done without sensible error owing to the minuteness of m, 

 though it could not be done in the foregoing values of dt and t, since the co-efficient 



4^2 in those values, is reciprocally great, as « is small) the quotient will be simply 

 u 



em a 



Substituting the value of u and dividing out sin a "we get the value of — in 

 the text. 





The integral of sin 2 p Utdt results from the formula T ainz ?)J?) = l(p 

 - sin 2<p, easily obtained by subatitutxDg for sb2 ip, its value --- cos 2?). 



4 



2 2 



