148 



The Rev. Samuel Haughton's Account of 



dAi _ dz dy dA^ _ dx 

 ~dr~^lt~ ^'dt' ~dr~^'di 



dz dA^ 

 dt' IF 



dy 

 di 



dx 

 It' 



Substituting in these equations the values of the velocities given by ( 1 ). we 

 obtain 



dA, „ R'--a' 



= -TO) ;; 



dt 



dA, 

 dt 



dA^ 

 dt 



a- 



X = (i?- -a-)p; 



P-^-y = {i^'-V)q. 



(12) 



= Pu 



R'- c" 



z = (R'- c') r. 



These equations prove, that the areolar velocity of the projection on a co-ordinate 

 plane varies at the ordinate to that plane. By means of the method of quadra- 

 tures, we may determine from equations (12) the position of the projections of 

 the principal axis at any instant, and hence deduce the position of the axis 

 itseE 



Second Method. , 



If the spherical conic be projected on a cyclic plane of the ellipsoid of gyra- 

 tion, bylines parallel to x and ^,the projections will be two concentric circles, and 

 the corresponding projections will lie on the same or- Fig. 2. 



dinate SII' (fig. 2). The inner circle will belong to 

 the projection parallel to x, if R be greater than h, 

 and will belong to the projection parallel to z if i? 

 be less than h ; and if ii be equal to h, the two circles 

 will coincide with each other and with the spherical 

 conic, which in this case becomes the circular section 

 of the ellipsoid. The projected point will revolve 

 round the circumference of the inner circle, and will vibrate on the circum- 

 ference of the outer circle, between the dotted lines. It is evident that the 

 mean axis of the ellipsoid OY lies in the plane of the figure, Let SI and SI' be 

 equal to p, p, and let C, (" denote the radii of the two circles : the velocities 

 of the projections in the circles will evidently be 



