ROTATION OF A RIGID BODY 149 



on the Continuous Displacements of the Particles of a Medium (Scientific 

 Papers, Vol. I, p. 37). But ere long he discovered a less artificial definition, 

 free from Cartesian symbolism. This mode of establishing the theory of 

 V is given in the appendix to his great paper already mentioned, that on 

 Green's and other Allied Theorems (1870, Sci. Pap. Vol. r, p. 136). Here 

 we find developed in an original manner the quaternion integrals through 

 volumes, over surfaces, and along edges, which include as special cases the 

 theorems of Green, Gauss, and Stokes. 



In 1868 Tait published an elaborate memoir on the Rotation of a Rigid 

 Body about a Fixed Point (Sci. Pap. Vol. i, p. 86), concerning which, while it 

 was in preparation, he had a good deal of correspondence with Cayley. On 

 August 1 8, 1868, he sent Cayley the concise quaternion equations of 15, 19, 

 21, and asked if the results are "merely a shortening of yours"; and on 

 October 17, the same year, he drew Cayley 's attention to the Cartesian formulae 

 in 28, 29, and to the fact that, " without integrating Euler's equations at 

 all (and I think from your second Report that the problem has always been 

 solved by first finding /, q, r), I find the following equations for w, x, y, z 

 [equation 24 in Paper]." On October 21, Cayley replied : 



" The rotation formulae are deducible by an easy transformation from formulae 

 in my paper (Cam. and Dub. Math. Journ. Vol. I, 1846).. ..But the actual form you 

 have given to the formulae is, so far as I am aware, new ; and a very decided improve- 

 ment as reducing the denominator to be of the third order." 



For these two quaternion papers Tait was awarded the Keith Prize by 

 the Royal Society of Edinburgh. The Secretary asked Clerk Maxwell to 

 draw up a statement to be read when the prize was formally awarded by the 

 President ; and Maxwell responded with a playful humour which considerably 

 mystified Professor J. H. Balfour. A copy was preserved in Tait's Scrap 

 Book. 



(Balfour, having asked Maxwell to write something which could be read at a 

 meeting of the R. S. E. when I was to get the Keith medal, was mystified as 

 follows. P. G. T.) 



GLENLAIR, DALBEATTIE, 28/11/70. 



Dear Professor Balfour, 



I do not presume to inform an officer of the Society with respect to 

 its recent awards. I saw that Tait had got the Keith Prize which is or ought to 

 be known to the public. I have not yet got a copy of the reasons for which it 

 was awarded, so if I coincide with them it does not arise from imitation. 



The question seems to be, What is Tait good for ? Now I think him good, 

 first, for writing a book on Quaternions, and for being himself a living example of 



