152 PETER GUTHRIE TAIT 



ensure its being + ve you must stick a sign to it, and that when you are proving 

 a minimum in certain cases the whole appearance of the proof should be trending 

 towards a maximum. 



" What do you recommend for El. and Mag. to say in such cases ? 



" Do you know Grassmann's Ausdehnungslehre ? Spottiswoode spoke of it in 

 Dublin as something above and beyond 4nions. I have not seen it, but Sir W. 

 Hamilton of Edinburgh used to say that the greater the extension the smaller the 

 intention." 



We have not the record of Tait's reply to the question of the sign, 

 a question which many later users of vector notations have attempted to 

 answer by simply ignoring one of the distinctive features of Hamilton's 

 calculus. So long as it is a question merely of a concise notation no harm 

 is done ; and Maxwell without seriously affecting the symbolic presentation 

 of his theory of electromagnetism might have adopted this method. But he 

 had too great a regard for the founder of Quaternions, and too deep an 

 insight into the inwardness of the quaternion calculus, to allow mere 

 expediency to play havoc with far-reaching fundamental principles. 



Meanwhile Tait's activity in developing quaternion applications continued 

 throughout the seventies. In a Note on Linear Differential Equations in 

 Quaternions (Proc. R. S. E. 1870; Set. Pap. Vol. i, p. 153) he struck out on 

 new paths. Here he gave an extremely simple solution of the problem 

 of extracting the square root of a strain or linear vector function. 



In a letter to Cayley of date Feb. 28, 1872, Tait gave the Cartesian 

 statement of the problem and continued, 



My quaternion investigation, which is very simple, leads to the biquadratic 



4m, 



where m, and m t are known functions of [the elements of the strain]; and from 6 

 the values of [the elements of the square root of the strain] can easily be found. 



Thomson and I wish to introduce this into the new edition of our first volume 

 on Natural Philosophy but he objects utterly to Quaternions, and neither of us 

 can profess to more than a very slight acquaintance with modern algebra so that 

 we are afraid of publishing something which you and Sylvester would smile at as 

 utterly antiquated if we gave our laborious solutions of these nine quadratic equations. 



As I said before the question is of interest in another way (for my Report on 

 Quaternions), for if <f> be the strain function and <' its conjugate, and if we try to 

 resolve, the strain into a pure strain followed by a rotation, so that <( ) = $&( )f~ l , 

 I find vt" ( ) = <f>'<f> ( ), so that the pure strain is the square root of the given strain 

 followed by its conjugate. 



