346 



arc. A very ready way of integrating these equations is to throw them 

 into the follo^ving somewhat different form : — 



Let {p) = perpendicular arc let fall from (P) on the great circle tan- 

 gent to the spherical curve whose running co-ordinates are (0) and {i^')', 

 then, by an easy application of iN'apier's rules for the solution of right- 

 angled spherical triangles, 



Sin p = sin ^9 . 



as 



equations (10 and (11) may be written 



^ = const = w sin (11) 



Sin jt? = ■ — -. — — (cos ^0 - cos 9) + sin 9^. (12) 

 w sm ^0 



Equation (12) answers to that of a curve in piano in terms of the radius 

 vector and the perpendicular on the tangent. The expression for the 

 radius of spherical curvature corresponding to the well-known formula 



dp 



is 



a cos 9 



[See Graves' translation of Chasles on " Cones and Spherical Conies."] 

 Applying this expression to the equation of the present curve, we 



get 



n 4. -D ^ z> ^ 1 ^ sin 6*0 



Cot ic = — : — OT IC- const = tan"^ ; 



to sm 9q m 



.-. the axis of the gyroscope describes a circular cone of a semi-angle 



tan ^ with an angular velocity . ^ .„ f -4t- 

 m sm E\ dt 



= + sin ' 



while the axis of the cone revolves round the polar line in a direction op- 

 posite to the earth's rotation with an angular velocity (w) ; in other 

 words, constantly points to the same fixed star. 



For completeness, I have thus solved the case where the axis is un- 

 constrained by the same methods as the other two. 



