VI 



calculus of plane curves : he initiates his discussions about geometry 

 of n dimensions : he founds the theory of invariants and covariants : 

 and he elucidates the connexion between doubly-infinite products and 

 elliptic functions. Some of these early papers are now classical ; and 

 the briefest inspection of them is sufficient to reveal the suggestiveness 

 and the easy strength of the young mathematician who was not yet 

 in his twenty-fifth year. 



Even by this date the opportunities of publication in England had 

 become inadequate to his needs. Curiously enough, he does not 

 appear to have sent any paper to the Royal Society until the year 

 1852, when Sylvester communicated the " Analytical Researches 

 connected with Steiner's Extension of Malfatti's Problem,"* to the 

 Society. Later in the same year, Cayley was elected a Fellow of the 

 Society, and thereafter many of his papers appear in its ' Transac- 

 tions.' Before 1852, there were few journals either at home or 

 abroad which did not receive communications from him : and even in 

 the quite early years of his researches, several of his papers, written 

 in French, appeared in Liouville's journal and in Crelle's journal. 

 As societies and journals grew in number, so the area over which his 

 papers spread became ever wider. 



At first, after winning his Trinity Fellowship, he remained at 

 Cambridge, and his time must then have been largely at his own 

 disposal. This freedom, in his circumstances, could last only for a 

 limited time, because, unless he either entered holy orders or devoted 

 himself to teaching in some permanent post (if obtainable) in the 

 College, the Fellowship could be held for not more than seven years 

 after his M.A. degree a period that would expire in 1852. He was 

 unwilling to take holy orders not that there was any religious 

 obstacle in his way, for he was not harassed either by philosophical 

 doubts or critical difficulties. His simple reason for remaining a 

 layman was that, though devout in spirit and an active Churchman, 

 he felt no vocation for the sacred office. 



In consequence, it became necessary to choose some profession. 

 Cayley selected the law, left Cambridge in 1846, entered at Lincoln's 

 Inn, and became a pupil of the famous conveyancer, Mr. Christie. A 

 story of their first interview, that Mr. Christie used to tell in after 

 years, is an illustration of the modesty and the lack of self-assertive- 

 ness which were leading features of Cayley's character : and this 

 impression is confirmed by the recollections of a fellow-pupil, Mr. 

 T. C. Wright, who says : 



" . . . . We fellow-pupils knew that Arthur Cayley had been 

 the Senior Wrangler of his year, and that he possessed extra- 



* Cayley's ' Collected Mathematical Papers,' vol. 2, No. 114. Subsequent refer- 

 ences to this series \vill be made in the form ' C. M. P.' 



