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greatest axis of the ellipsoid, and next by right lines parallel 

 to its least axis, each projection of the conic will be a circle 

 having its centre at O. The two projections P and Q of the 

 same point I will always be in a right line perpendicular to the 

 mean axis ; let this right line cut the mean axis in M. Then, 

 while the point I describes the spherical conic, the points P 

 and Q will move in their respective circles, in such a way, 

 that the velocity of P will vary as the ordinate MQ, and the 

 velocity of Q will vary as the ordinate MP. 



From this theorem we immediately obtain the elliptic in- 

 tegral which represents the time. For, supposing the greater 

 circle to be that described by Q, and taking its radius for unity, 

 if we put c for the radius of the other circle, and denote by ^ 

 the complement of the angle which OP, in any position of 

 P, makes with the mean axis of the ellipsoid, we have 

 MQ = VI — cHm^^, and therefore 



dt- 



\/ 1 — 6'^siu^^ 



where dt is the element of the time, and A is a constant quan- 

 tity. Hence, the time at which the point P attains any given 

 position in its circle, and therefore the time at which the point 

 I attains any given position in the spherical conic which it 

 describes, is determined by an elliptic function of the first kind, 

 the modulus and amplitude of which are exhibited geometri- 

 cally. The modulus c of the function is obviously the ratio 

 of the two moduli of the cone which the right line 01 describes 

 within the body ; for the radius of each circle is found by 

 dividing the distance 01 by one of the moduli of the cone. 



The preceding method of determining the time in the pro- 

 blem of rotation occurred to the author in the year 1831, and 

 has since been given at his lectures in Trinity College. 



It may be observed, that the motion of the axis of rotation 

 within the body is known when that of the right line 01 is 

 known ; for that axis is always perpendicular to the plane 

 which touches the ellipsoid in the point I. 



