120 PETER GUTHRIE TAIT 



account of the time I should have had to spend in acquiring a sufficient knowledge 

 of them. 



I have all along preferred mixed, to pure, mathematics, and since I left 



Cambridge, where the former are little attended to, have been busy at the Theories 



of Heat, Electricity, etc. Your remarkable formula for = \- = \- ^- as the square 



da? dy* dz* 



of a vector form, and various analogous ones with quaternion operators, appear to 

 me to offer the very instrument I see*, for some general investigations in Potentials, 

 and it is therefore almost entirely on the subject of Differentials of Quaternions that 

 I shall trespass on your kindness.... 



The correspondence thus begun continued week by week with wonderful 

 continuity until July 1859, when Hamilton began to print the Elements. 

 The successive letters were numbered (Hamilton's in Roman, and Tail's in 

 Indian, numerals) and copies kept by the writers themselves, so that there 

 might be no difficulty in referring to questions raised by either at all stages 

 of the correspondence. 



In his letter of August 20, 1858, Tait mentioned particularly certain 

 difficulties : 



Perhaps it is only due to the novelty of the subject, but I have felt at several 

 points that the otherwise known result was (perhaps not necessary but at all events) 

 very desirable, in suggesting the transformation suitable for its proof. As instances 

 I may mention * found in Art. 474 of your Lectures for the value of 



p* + 4 (t )* Sip Step, 



and the transformation of the Tractor function for the 2nd integration of the 

 equation of motion of a planet.... 



Again in Art. 591 I cannot see how you infer that v is a normal vector when 

 the equation to a surface is put in the form Svdp = o, Tdp not being indefinitely 

 small, because it seems to me that in such a case v is a vector perpendicular to the 

 chord dp. 



It was in reply to Tail's difficulties regarding the notion of finite 

 differentials that Hamilton wrote the long letter v, which might have been 

 a chapter in a treatise on the fundamental conception of the fluxion or 

 differential method. Hamilton subsequently gave the argument clearly in 

 his second treatise, the Elements of Quaternions, developing the whole 

 discussion from the definition : 



Simultaneous Differentials (or Corresponding Fluxions) are limits of equimultiples 

 of simultaneous and decreasing Differences. 



In this remarkable letter (dated October n to October 16, 1858) which 

 occupies 45 closely written pages of large-sized note paper, and is subdivided 



