ORTHOGONAL ISOTHERMAL SURFACES 153 



Cayley replied, March 2, 1872: 



" I find that your question may be solved very simply by means of a theorem in 

 my memoir on Matrices, Phil. Trans. 1858." 



He then proceeded to indicate the steps of a somewhat prolonged 

 process by which the solution might be found ; but in a second letter 

 written a few hours later he practically reproduced Tail's process by use 

 of the symbolic cubic, the Matrix symbol M being written instead of the 

 vector function <f>. 



On March 5, Tail wrote: 



It is a most singular fact that you seem to have been working simultaneously 

 with Hamilton in 1857-8, just as I found you had been in a very much earlier year ...... 



I have had but time for a hurried glance at your paper on Matrices and I see that 

 it contains (of course in a very different form) many of Hamilton's properties of the 

 linear and vector function....! send you a private copy of my little article, by which 



you will see how closely the adoption of Hamilton's method has led me to anticipate 

 almost every line of your last note.... There is one point of Hamilton's theory to which 



I do not see anything analogous in your paper. Expressed in his notation it is that 



- 1 and 

 are identical, if we have gh mSfT^^r^p. 



The Report referred to by Tait in his letter of Feb. 28 was a Report 

 which, urged by Cayley, he had agreed to prepare for the British Associa- 

 tion. Shortly afterwards he asked to be relieved of the task, as it would 

 be of too personal a character, and suggested Clifford as eminently quali- 

 fied to undertake it. Nothing further seems to have been done. 



The quaternion discussion of orthogonal isothermal surfaces was 

 published in 1873 (Sci. Pap. Vol. i, p. 176). It is an interesting example of 

 the use of Hamilton's rotational operator g( )^ -1 . The opening paragraphs 

 of this paper are not quaternionic, and seem to have been introduced by 

 Tait for the double purpose of showing how he originally began to attack 

 the problem and how much more suggestive and concise the quaternion 

 solution is. In a letter of date July 22, 1873, Maxwell referred in a 

 deliciously humorous manner to the character of Tail's investigations in 

 these words : 



" I beg leave to report that I consider the first two pages of Professor Tait's Paper 

 on Orthogonal Isothermal Surfaces as deserving and requiring to be printed in the 

 Transactions of the R. S. E. as a rare and valuable example of the manner of that 

 Master in his Middle or Transition Period, previous to that remarkable condensation 



T. 20 



