138 PETER GUTHRIE TAIT 



When I came to your equation (31) of xxin I tried to prove it for myself 

 and was so successful that I was just about to send you a note on the subject when 

 I luckily read on and found that your luminous thought had completely anticipated 

 me. Here is my work as it stands in an MSS book. 



Change <f> into $+g, &c. and multiply by M, 



or 



No letter from Hamilton of date later than July 19, 1859, has been 

 preserved, although there are copies of eight of Tail's own letters to 

 Hamilton ranging from Sept. 7, 1859, to January 14, 1861. From these 

 we gather that Hamilton was absorbed in the preparation of his new book 

 and was keeping Tait steadily supplied with the proof sheets of the earlier 

 chapters. Meanwhile Tait was strengthening himself in the use of the 

 calculus, and in letter 41 of date Sept. 26 gave, very much as it afterwards 

 appeared in his Treatise, his quaternion investigation of Ampere's electro- 

 dynamic theory. This investigation, especially in the more generalised form 

 in which it was presented in his paper of 1873 on the various possible 

 expressions for mutual forces of elements of linear conductors (Proc. R. S. E. 

 vin ; Set. Pap. Vol. i, p. 237), is a good example of the directness with 

 which the quaternion method deals with a general problem 1 . Beginning 

 with a general form of function, involving the relative position and the 

 directions of two current elements, Tait developed the form of this function 

 by a skilful use of Ampere's fundamental experimental laws. In letters 42 

 and 43 of date Nov. 3, 1859, and March 22, 1860, Tait continued the 

 development of his electrodynamic investigations, pointing out the importance 

 of the vector 



Vaa! fdUa 



in all investigations connected with the action of a circuit, where a! is the 

 element at the point a of the circuit. 



A few months later Tait commenced his Edinburgh career, having 

 been helped thereto by the following testimonial from Hamilton : 



Understanding that Professor Peter Guthrie Tait, now of the Queen's College, 

 Belfast, but formerly of St Peter's, Cambridge, is likely to become a candidate for 



1 See also Clerk Maxwell's Electricity and Magnetism, Vol. n, Chap. n. 



