154 PETER GUTHRIE TAIT 



not to say coagulation of his style, which has rendered it impenetrable to all but 

 the piercing intellect of the author in his best moments." 



When this paper was passing through the press Tait had a brief 

 correspondence with Cayley on the nature of his solution. After its 

 publication, Cayley made some interesting comments in a letter of date 

 March 25, 1874. He first reproduced one of his own results which shows 

 that, in order that r = const, may represent a family of orthogonal surfaces, 

 then r considered as a function of x y z must satisfy a somewhat complicated 

 partial differential equation of the third order. Tail's equation da- = uqdpq' 1 , 

 he then pointed out, must be the equivalent of this partial differential equation 

 of the third order. He concluded in these words : 



"Do you know anything as to the solution when the limitations [imposed by 

 Tait] are rejected, and imaginary solutions taken account of? Considering simply 

 the equation of the third order and the equation a + 6 + c = o [that is W=o] it would 

 seem probable that there must be a solution of greater generality than the confocal 

 quadrics. I do not see my way to the discussion of the question. The condition 

 a + & + c=O seems to make no appreciable simplification in the equation of the third 

 order. I admire the equation da = uqdpq~ l extremely it is a grand example of the 

 pocket map." 



This comparison of a quaternion formula to a pocket map was quite 

 in accord with Cayley's attitude towards the quaternion calculus. He 

 admitted the conciseness of its formulae, but maintained that they were 

 like pocket maps : everything was there, but it had to be unfolded into 

 Cartesian or quantic form before it could be made use of, or even understood. 

 This view Tait combated with all the skill at his command ; and every now 

 and again the two mathematicians had a friendly skirmish over the relative 

 merits of quaternions and coordinates. 



Even when they exchanged views on quaternionic problems altogether 

 apart from this central controversial question, their different mental attitude 

 came clearly to the front in their correspondence. This is seen, for example, 

 in the following series of letters. 

 Dear Tait 



In the quaternion q = w + ix+jy + kz, assuming 



tan -/= + - y ' + ** , (r = V(* + j>' + *') and -, y -, - , = cos a, cos & cos 7) 



then the quaternion is 



q *=w + ix+jy + kz 



* 



= - i' > {cos \f+ sin \f(i cos a +j cos /3 + k cos 7)} 



sin ^T 



and we can interpret the quaternion in a twofold manner, viz., in the first form, 



