XX111 



that, ultimately in the development, considerable generalisations are 

 effected. Now the usual writer, in publishing the results of such an 

 enquiry, draws them up in a sequence that partly marks the order of 

 their connected discovery : and, in doing so, he makes his work easier 

 for his readers. But Cayley was not the usual writer. When he had 

 reached his most advanced generalisations he proceeded to establish 

 them directly by some method or other, though he seldom gave the 

 clue by which they had first been obtained : a proceeding which does 

 not tend to make his papers easy reading. An instance of the fact 

 occurs* in his 'Memoir on the Theory of Matrices,' where lie proves 

 that a matrix satisBes an algebraical equation of its own order; he 

 proves it by verification in simple cases, but he gives no clue as to his' 

 line of discovery. An instance of the method occurs in a notef added 

 to one of his papers, where he says that the general equations 



{yd x } yd x = 0, {xd y } xd y = 0, 



characteristic of co variants and invariants of binary quantics were 

 initially suggested by considering the relation of the quadratic 

 aa? + 2bxy + cy* and its discriminant ac b 2 to these equations. In the 

 paper he drops linear transformation as connected with the covarian- 

 tive property and defines a covariant as a function satisfying these 

 two equations. 



His literary style is direct, simple and clear. His legal training 

 had an influence, not merely upon his mode of arrangement but also 

 upon his expression ; the result is that his papers are severe and 

 present a curious contrast to the luxuriant enthusiasm which pervades 

 so many of Sylvester's papers. He used to prepare his work for 

 publication as soon as he had carried his investigations in any subject 

 far enough for his immediate purpose. He found it an easy matter 

 to do this part of his work, and thus differed widely in experience from 

 those to whom the preparation of a paper is laborious even when the 

 results to be incorporated have been obtained. As a matter of fact, he 

 took the straightforward course of saying what he had to say in a clear 

 and simple manner, fixing his mind upon the substance and never going 

 out of his way in order to secure beautiful form for the presentation 

 of results. Yet not infrequently his papers are so admirably written 

 that they satisfy the exacting critics ; thus it is perhaps not too much 

 to affirm that his ' Sixth Memoir on Quantics 'J could not be presented 

 in more attractive form a character due, however, to the tendency 

 of his method and to his results, but not acquired by any effort 

 specially devoted to elaboration of clear expression. Again, a paper 

 once written out was promptly sent for publication ; this practice he 



* ' 0. M. P.,' vol. 2, No. 152, pp. 482, 483 ; ' Phil Trans.' (1858), pp. 24, 25. 



t ' C. M. P.,' vol. 2, p. 600. 



t ' C. M. P.,' vol. 2, No. 158; ' Phil. Trans.' (1859), pp. 6190. 



