Notices respecting New Books. 419 



One chief interest peculiar to the new edition is the Appendix 

 of Thirteen Notes covering 111 pages of the Second Volume. 

 These are the work of Professor Joly, and are intended to show 

 the various directions along which Hamilton's powerful calculus 

 has been or may be suitably extended. They include such subjects 

 as the theory of screws, finite rotations, the kinematical treatment 

 of curves and surfaces, systems of rays, and in particular important 

 sections on the linear vector function and on the differential 

 operator V- 



The further development of the theory of the linear vector 

 function and its application to strains and stresses constitute part 

 of Tait's important additions to Hamilton's own labours ; and the 

 fact is duly acknowledged by Professor Joly. Bat by far the most 

 important of Tait's quaternion investigations have to do with the 

 operator V- Hamilton discovered it ; but it was left to Tait to 

 disclose its full potency and show how to use it. It is not a 

 little surprising, therefore, that Professor Joly should not give 

 the remotest hint that this whole department of quaternions was 

 virtually created by Tait, whom Maxwell named "the chief 

 Musician upon Nabla"*. 



When, in 1870, Tait was awarded the Keith Prize by the Eoyal 

 Society of Edinburgh, Maxwell, referring to the papers for which 

 the award was given, wrote in November of that year : — " That on 

 Eolation is very powerful, but the last one on Green's and other 

 allied theorems is really great "f . 



In the same communication Maxwell says further: " No one can 

 tell whether he (Tait) may not yet be able to cause the Quaternion 

 ideas to overflow all their mathematical symbols and to become 

 embodied in ordinary language so as to give their form to tho 

 thoughts of all mankind. I look forward to the time when the 

 idea of the relation of two vectors will be as familiar to the 

 popular mind as the rule of three, and when the fact that »"/= — ji 

 will be introduced into hustings speeches as a telling illustration." 

 It was, no doubt, with this prophetic thought in his mind that 

 Maxwell introduced the expressive and compact notation of 

 quaternions into his 'Electricity and Magnetism.' In these days 

 Maxwell's example is followed by many workers in electro- 

 magnetic theory. Most of them are no doubt content to use 

 merely as a notation the particular form of vector analysis which 

 they adopt ; and with the rectangular space scaffolding ever before 

 their eyes they spoil all hope of real quaternionic work by discarding 

 the associative principle which makes quaternions workable, and 

 without which a vector analysis speedily becomes too complicated. 

 11 I can work Heaviside's methods but I don't understand 



* See Maxwell's Life, p. 634. The name Nabla, which Professor Joly 

 introduces almost as if Hamilton himself had given it, was suggested by 

 W. Robertson Smith from the resemblance of the symbol v to an Assyrian 

 harp. See Tait, " On the Importance of Quaternions in Physics," Phil. Mag. 

 vol. xxix. p. 91 ; Scientific Papers, vol. ii. p. 303. 



t Jn his ' Electricity and Magnetism,' the same paper is characterized as 

 " very valuable." 



