20 



THE PENDULUM AND GYROSCOPE 



^=n sin Z±J'^- 



Ropresenting tlicsc two valiu-s by /3i and /^s, equations (5) may be put in the 

 form (l)y writing for 1), and E„ C\ cos e^ and C\ sin £„ &c., and reducing) 

 x= C, cos (/3i<— fi)+ f 2 cos (/32^— fa) 

 y^ — C, sin (/3i<— fi)— t'o sin {(5^1— s^) 



Assumino- for /=0, a— and ';'^^0, it will give fi = f2=l 7t, and the above 



become 



x=(7i sin /^i^+Tz sin /?2< 

 y=Ci cos[iil-\-C\ cos/ia^ 



Instead of the arbitrary constants C^ and (X we may write \{A^B) and |(^— 5), 

 at the same time substituting the values of ^^ and /?3 and developing; by which 

 the preceding equations become (putting n sin ?t=?i') 



a--=J. cos p. i! sin ti! t-\-B sin P- ;; cos ir! t 



(«) .-^ .- 



?/=A cos p < cos n't—B sin S' y gin «'^ 



If we transfer the co-ordinates now referring to (relatively) fixed axes, to others 

 moving with the relative angular velocity w', that is, if we transfer to axes making 

 at any instant the angle h. sin ;i< with the fixed ones, the new co-ordinates will 

 have the values 



x'=^x cos n't — y sin id 



y'=x sin n'i-\-y cos h'< 



or, substituting values of x and y, 



w 



x'^B sin I 



t 

 I 



y'^A cos K- 1 



J'roin which we may obtain 



• A^ x'^-\-B- y'^^A' B' 



which is the equation of an ellipse, having A and B for semi-transverse and semi- 

 conjugate axes. If i^=() the ellipse becomes a right line, hence the earth's rota- 

 tion causes an azimuthal motion of this line, or of the axes of the ellipse if the 

 motion is elliptical, equal to the component of that rotation about the local axis 

 and in the reverse direction. 



The motions of the "gyroscope pendulum," which is but the ordinary gyroscope 

 with an exceedingly long arm (or distance y, of my analysis, from the point of 

 support m the axis to the centre of gravity), are indicated "by equations precisely 

 similar to the above, deduced from an identical analysis; always assuming, as in 

 the solutions just given, that the arcs of vibration are small, so that vertical motions 



