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VIII. — On the Rotation of a Rigid Body about a Fixed Point. By Professor Tait 



(Received October 13th, Eead December 21st, 1868.) 



Although it is very improbable that there remains to be discovered any new, 

 and at the same time simple, fact connected with a question which has been 

 elaborately treated by many of the greatest mathematicians of this and the pre- 

 ceding century, the employment of a new mathematical method may enable us 

 to present some of their results in a more intelligible form, and with far less 

 expenditure of analytical power than has hitherto been deemed necessary ; and 

 it may give us such an insight into the question, that we shall be able easily to 

 discover the mutual relations among the various processes which have been 

 already employed; so far, at least, as these differ in principle, and not merely in the 

 peculiar co-ordinates assumed for the purpose of simplifying the equations. Such 

 a method is that of Quaternions, which seems to be expressly fitted for the 

 symmetrical evolution of truths which are usually obtained by the ordinary Car- 

 tesian methods only after great labour of calculation, and by modes of attack so 

 indirect, and at first sight so purposeless, as to bewilder all but a very small 

 class of readers. Quaternions afford so clear a view of the nature of the question 

 they are applied to, that even the student, if he have some little knowledge of 

 them, can often see why a transformation is made, whose object he would have 

 been unable to discover had the problem been masked in the unnecessarily arti- 

 ficial difficulties of Cartesian geometry, or the outrageously repulsive formulae of 

 spherical trigonometry. 



By far the most elegant and most easily intelligible representations of the 

 motion of a solid body yet discovered, are due to Poinsot. With the following 

 extract from his splendid work, Theorie Nouvelle de la Rotation des Corps (Liou- 

 vilWs Journal, 1851), I most cordially agree, — though it appears to me that, when 

 he does condescend to use analytical methods, he is by no means so happy as 

 others have been, who, trusting to mathematical analysis alone, had not the 

 benefit of his beautiful geometrical representations. But in perusing the extract, 

 let the reader bear in mind that a quaternion equation is quite as suggestively in- 

 telligible, to those who understand it, as any geometrical diagram can possibly be. 

 In fact, I might almost say, that it is more readily intelligible than diagrams usually 



VOL. XXV. PART II. 3 X 



