of a rigid Body round ajixed Point. 1 5 



come, the one logarithmic, the other circular ; as may be thus 

 shown : 



In the general equation (20.) let u = b, then we find 



Now let {a^--c^) z" = c^ (a^-b^) sin^ <^, (31.) 



where <p, as may be shown, is the angle between u and the 

 mean axe of the ellipsoid, measured on a circular section of 

 the surface ; by this transformation, equation (30.) is changed 

 into 



dt = ■ ■ ^ , (S2.) 



f ^{a^-b^) {b^-c') smc;>' ^ ' 



or, t = ;; ===== log tan — — \- constant . . (33.) 



f V{a^-b^){b''-cf S 2 ^ ^ ' 



Hence if the axis u is found in one of the circular sections of 

 the ellipsoid, at the commencement of the motion, the plane 

 of the principal section of this surface containing the semi- 

 axes a and c, will indefinitely approach to, yet never actually 

 coincide with, the plane of the impressed moment. 



XXIV. The angle ^ in this case may be determined thus : 

 In the integral of (26.), namely, 



putting b for u, and for dt and z their values given by (32.), 

 (31.), the last equation is transformed into 



^ wb ^ , fCj^7^~W ■^ , , 



—■Or =-s!t J 7^ tan~* < - — . ^ cos A y + constant (34.) 



J \,avb^ — c^ J 



XXV. When two of the principal moments of inertia are 

 equal, as A = B, the ellipsoid becomes a spheroid of revo- 

 lution, and the time with the angle i/r are determined by cir- 

 cular arcs. 



XXVI. The axis oftlie rotatory motion caused by the cen- 

 trifugal forces^ lies in the -plane of the impressed moment. 



The cosines of the angles which the tangent at the vertex 

 of u, to the conic section whose semiaxes are u and vt, (see 



, d X 



fig. in art. XI.) makes with the axes of coordinates are-^ — , 



— =^, — — ; hence these differential coefficients represent the 

 ds ds 



cosines of the angles, which ?« parallel to u makes with the 



same axes. 



