i68 PETER GUTHRIE TAIT 



ever be the principle of notation adopted, whether a modified Hamiltonian 

 or Grassmannian, these systems are used as Maxwell used quaternions. 

 Tait inspired Maxwell to use the quaternion vector symbolism. All vector 

 analysts follow Maxwell in substituting for the diffuse Cartesian symbolism 

 a more compact and graphic vector notation. In imitating Maxwell they 

 become disciples of Tait : and once more we realise the close historic 

 connection between Tait's quaternion labours and the developments of 

 modern vector analyses applied to physical problems. 



It was indeed for the sake of physical applications that Tait made 

 himself master of the quaternion calculus. The conditions under which 

 his Elementary Treatise* was prepared have already been described ; and 

 in judging of the merits of the book, especially in its first edition, we 

 must bear in mind that Hamilton's expressed wishes considerably tied 

 Tait's hands. It was necessary for the sake of the student that the 

 foundations of the calculus should be established in one of the several 

 ways which Hamilton himself had already indicated. But Tait's aim, as 

 indicated in the Preface to his Treatise, was to bring out the value of 

 Quaternions in physical investigations. 



In the earlier chapters (I refer at present only to the first edition of 

 the Treatise) there was of course little scope for Tait to show any originality 

 of treatment. The first chapter in which he began, as it were, to beat 

 out his own path, was Chapter V, on the solution of equations. In 

 discussing the properties of the linear vector function he followed a line 

 suggested by Hamilton in one of his letters ; but he followed it out in his 

 own way. In Tait's eyes the linear vector function was a strain ; and to 

 a reader acquainted with the theory of strains it is abundantly evident 

 that, even when explicitly confining himself to the purely mathematical 

 side of the question, Tait had the strain conception vividly before his 

 mind. In this early chapter he emphasised those properties which became 

 all important in the later chapters on Kinematics and Physics. 



The linear vector function continued to occupy Tait's attention through- 

 out the remaining years of his life ; and many interesting applications 

 were added in the second and third editions of his book. These usually 

 appeared, in the first instance, as notes to the Royal Society of Edinburgh. 

 He never found time to put his investigations into the form of a complete 

 memoir. All he was able to accomplish was a series of abstracts giving 

 1 First edition, 1867; and edition, 1873; 3rd edition, 1890. 



