i 3 o PETER GUTHRIE TAIT 



and on just now glancing at your No. 22 received yesterday or the day before, but 

 quite unexamined hitherto...! see that the symbol 



occurs several times. You have therefore probably introduced some new definition 

 of the functional symbol and I am not entitled tc>6ay that your formula requires 

 any correction. Of course we cannot afford to part with a certain liberty, of notation. 

 But with my meaning of <f> as developed in my Lectures and Letters, I found, a 

 few minutes ago the hint (as I admit) having been taken from your last letter 

 that the formula, 



{ 7X*- 1 - p')-* p}' = - Sp (P 3 - r 1 )- 1 P, 



is an identity ; and therefore that one of my symbolical forms of the equation of the 

 wave, namely, the equation 



may be immediately transformed to the following 



a result which I confess that I had not expected, but which (I suppose) agrees 

 substantially with yours ____ You deserve I think great credit for having percei 'ved this 

 transformation...." 



Thus we owe to Tait the discovery that the square root of a linear 

 vector function or matrix of the third order enters symbolically into certain 

 expressions exactly like an ordinary algebraic quantity. He was led to this 

 discovery by a comparison of his own special notation with the notation used 

 by Hamilton, who, on his own confession, had never thought of treating 

 the linear vector function in this way. It is not a little curious that, at 

 the time, neither Hamilton nor Tait seemed to have considered the analytical 

 significance of the square root of a linear vector function. This was done 

 in 1870 by Tait whose results, based on kinematic considerations, led to 

 an interesting correspondence with Cayley and a further development of 

 the properties of the matrix (see below, p. 152). 



After a good deal of further correspondence on the subject of the 

 Wave Surface, Hamilton communicated his method to the Royal Irish 

 Academy, and Tait published his investigation in the Quarterly Journal 

 of Mathematics. Meanwhile, in Hamilton's mind a new project had been 

 forming itself, which was first referred to in paragraph 71 of letter xiv, 

 written on January 21, 1859. Here Hamilton wrote: 



" [71] I must tell you however of a quite different project of mine, which may 

 occupy a good part of the present year if a fair share of health is spared me. 

 I want to prepare for 1860 though I do not forget a passage in St James either 

 a new edition of my Lectures, or what may be better, an entirely new work, which 



