188 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



limits the two values + JTT. This case may be obtained by putting 

 k = 1 in (7), or a = TT in (11) ; and we find 



= co cos 6 ................................. (15), 



^=logtan(i7r+|(9) .................. (16), 



x = - log tan (ITT + J0) - - sin 0, 





See the curve II of the figure*. 



It is to be observed that the above investigation is not restricted 

 to the case of a solid of revolution ; it applies equally well to the 

 case of a body with two perpendicular planes of symmetry, moving 

 parallel to one of these planes, provided the origin be properly 

 chosen. If the plane in question be that of xy, then on transferring 

 the origin to the point (M"/B, 0, 0) the last term in the formula 

 (1) of Art. 123 disappears, and the equations of motion take the 

 form (2) above. On the other hand, if the motion be parallel to 

 zx we must transfer the origin to the point ( N'/C, 0, 0). 



The results of this Article, with the accompanying diagrams, 

 serve to exemplify the statements made near the end of Art. 121. 

 Thus the curve IV illustrates, with exaggerated amplitude, the 

 case of a slightly disturbed stable steady motion parallel to an 

 axis of permanent translation. The case of a slightly disturbed 

 unstable steady motion would be represented by a curve con- 

 tiguous to II, on one side or the other, according to the nature of 

 the disturbance. 



125. The mere question of the stability of the motion of a 

 body parallel to an axis of symmetry may of course be more simply 

 treated by approximate methods. Thus, in the case of a body 



* In order to bring out the peculiar features of the motion, the curves have 

 been drawn for the somewhat extreme case of A =5B. In the case of an infinitely 

 thin disk, without inertia of its own, we should have A/B = cc-, the curves would 

 then have cusps where they meet the axis of y. It appears from (5) that i has 

 always the same sign, so that loops cannot occur in any case. 



In the various cases figured the body is projected always with the same impulse, 

 but with different degrees of rotation. In the curve I, the maximum angular 

 velocity is J2 times what it is in the critical case II ; whilst the curves III and 

 IV represent oscillations of amplitude 45 and 18 respectively. 



