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LVIII. On the flotation of a Rigid Body about a Fixed Point. 

 By J. J. Sylvester, M.A., F.R.S.* 



IN the Cambridge and Dublin Mathematical Journal for 

 March 1848, an article by Professor Stokes, of the Univer- 

 sity of Cambridge, is ushered in with the words following: — 



"The most general instantaneous \ motion of a rigid body 

 moveable in all directions about a fixed point consists in a 

 motion of rotation about an axis passing through that point. 

 This elementary proposition is sometimes assumed as self- 

 evident, and sometimes deduced as the result of an analytical 

 process. It ought hardly perhaps to be assumed, but it does 

 not seem desirable to refer to a long algebraical process for 

 the demonstration of a theorem so simple. Yet I am not 

 aware of a geometrical proof anywhere published which might 

 be referred to." 



The learned and ingenious professor is indubitably right, 

 and might have trusted himself to assert less hesitatingly the 

 necessity of demonstrating this proposition, which possesses 

 none of the characters of a self-evident truth ; but it is to be 

 regretted that he should have stated it in such a form as na- 

 turally to lead the incautious reader to mistake the nature 

 and grounds of its existence, which consist in this fact — that 

 any kind of displacement of a body moveable about a fixed 

 axis, whether instantaneous and infinitesimal, or secular and 

 finite, is capable of being effected by a single rotation about a 

 single axis. 



The annexed simple proof of this capital law has the ad- 

 vantage of affording a rule for compounding into one any two 

 (and therefore any number of) rotations given in direction, 

 magnitude and order of succession. . 



It will somewhat conduce to simplicity if we fix our atten- 

 tion upon a spherical surface rigidly connected with the rota- 

 ting body, and having its centre at the fixed point thereof. 

 When the positions of two points in this are given, the posi- 

 tion of the body is completely determined. 



Now evidently two points A, B may be brought respectively 

 to A' B f (if AB = A'B') by two rotations ; the first taking place 

 about a pole situated anywhere in the great circle bending at 

 right angles A A', the second about A', the position into which 

 it is brought by the first rotation. This view leads us to con- 

 sider the effect of two rotations taking place successively about 

 two axes fixed in the rotating body. Or again, we may make 

 the plane A' B' revolve into the position AB round a pole 



* Communicated by the Author. 



t The italics do not exist in the original. 



