CORRESPONDENCE WITH CAYLEY 161 



I showed the R.S.E., on Monday last, an Ellipse and a Hyperbola separately 

 tracing the same glissette. The uninitiated were much puzzled to see it, as the one 

 curve merely oscillates while the other turns complete summersalts, and they could 

 not conceive that the same curve could be traced by a point of each. But it comes 

 merely to this: that [in the parallelogram linkage OABA' which was sketched in 

 the letter] the ellipse describes B about O virtually by the two links OA, AB ; while 

 the hyperbola does it by the other two sides of the parallelogram. The centre, A, 

 of the ellipse has a to and fro motion through a limited angle, while A' (the centre 

 of the hyperbola) goes completely round. 



A later letter from Tait gave a further investigation of this problem 

 very much as it appeared in the published paper (Proc. R. S. E. Dec. 1889; 

 Sci. Pap. Vol. ii, p. 309), which Cayley characterised as " very interesting." 



On January 24, 1890, after acknowledging the receipt of Tait's 

 Quaternions and a copy of the Phil. Mag. paper on the Importance of 

 Quaternions in Physics (Sci. Pap. Vol. n, p. 297), Cayley renewed the old 

 discussion in these words : 



" Of course I receive under protest ALL your utterances in regard to coordinates. 

 Really, I might as well say, in analytical geometry we represent the equation of a 

 surface of the second order by 7=o; compare this with the cumbrous and highly 

 artificial quaternion notation Sp<j>p = l. But you cannot contend that this last 

 equation by itself contains the specification of the constants which determine the 

 particular quadric surface ; and the fair parallel is between your quaternion equation 

 and (*$*, y, z, i) J = o: and if you say yours is shortest, I should reply, mere shortness 

 is no object, or again there is nothing easier than to use a single letter to denote 

 (x, y, 2, i). Again, for a determinant 



y z 



X 1 y' z 



x" y" z 



there is here absolutely nothing superfluous, the determinant depends upon nine 

 quantities which have to be specified : and these are not simply a set of nine, but 

 they group themselves in two different ways into 3's as shown by the lines and 

 columns." 



Tait replied as follows : 



38 GEORGE SQUARE, 

 EDINBURGH, 



25/1/90. 

 My dear Cayley 



I might say with a great rhetorist, " I am not careful to answer thee in 

 this matter": but I think that most of your remarks seem to be based on ignoration 

 of the Title of my little paper. It is the use in Physics that I am speaking of. 

 i/=o is just as expressive in quaternions as in any other calculus, i.e. it is, in all, 



T. 31 



