RIGID BODY ABOUT A FIXED POINT. 263 



its angular velocities about its principal axes are given, or can be found. It was 

 not till after my investigations were nearly completed, and the chief fundamental 

 equations had been communicated to the British Association at Norwich, that I 

 became aware of the existence of Professor Cayley's* admirable Second Report on 

 Theoretical Dynamics, which contains an immense amount of valuable informa- 

 tion, especially bearing on the present subject. From this I found that the 

 notion of attaining symmetry, by seeking the single rotation which would bring 

 the body from some initial position to its actual position at a given time, which 

 had been suggested to me by Hamilton's! beautiful results, is due to Euler; 

 and I also found that, by the help of certain formulae due to Rodrigues, Cayley 

 has completely solved the question in the " Cambridge Mathematical Journal, 1 ' 

 vol. iii. (1843). | Comparative symmetry, however, is only attained by means of 

 a brilliant display of analytical power at a great expense of time and bewilder- 

 ment to the ordinary reader. In the " Philosophical Magazine," 1848, ii., Cay- 

 ley has translated some of his formulae into quaternions, and has thus arrived, 

 though by a very circuitous route, at the fundamental kinematical equation of 

 the present paper (§7 below). He does not give it in its simplest form, and he 

 remarks that he has " not ascertained whether it leads to any results of import- 

 ance." Under these circumstances, I have had no hesitation in laying this 

 paper before the Society ; for although many of its more important results have 

 been otherwise obtained, few, with the exception of those due to Hamilton 

 (which will be given in their turn), have hitherto been arrived at so easily or in 

 such simple forms. 



As symmetry has been the particular object which I have had in view, by far 

 the greater part of the investigation bears upon the determination of the qua- 

 ternion, by which the transition can at one step be effected from any initial 

 position to the actual position of the body at a given time ; and a good many 

 results have been retained, which are of more interest as properties of quater- 

 nions, than as regards their connection with the physical question. In the kine- 

 matical part of the paper, to which I proceed as a necessary preliminary, I have 

 exhibited, for facility of comparison with other works on the subject, the values 

 of this quaternion in terms of the various sets of co-ordinates usually employed. 

 This, I need hardly say, does not lead to very simple or elegant results ; but the 

 fault is due, not to quaternions, but to the unnaturalness and want of symmetry 

 of these common methods of attacking the problem. On the other hand, nothing 

 can be neater than the set of formulae which are suggested directly by quaternions. 



* Beport on the Progress of the Solution of certain Special Problems of Dynamics. — Brit. 

 Ass. Report, 1862. 



t Proc. B. I. A., 1846. See also §§ 1 and 4 below. 



\ See also Cambridge and Dublin Math. Journal, vol i. (1846). 



