XXiV 



maintained throughout his life. He undoubtedly formed projects 

 for the immediate future; thus to the second edition* of his 'Treatise 

 on Elliptic Functions ' he intended to add a couple of chapters which, 

 however, remained unwritten solely for the reason that all such pro- 

 jects were carried into effect only about the time when the need arose. 

 The consequence is that he has left few arrears of unfinished or un- 

 published papers ; his work has been given by himself to the world. 



Only one other remark as to the form of his papers need be made. 

 Readers must be struck with the number of exact references he makes 

 to other writers. It was a practice about which he had very decided 

 opinions : he wished not merely to make honourable acknowledgment 

 of indebtedness but also to give indications of the history of the sub- 

 ject. In the latter particular he was always careful to insert in the 

 reference the year in which the book or the paper had appeared ; and 

 he steadily urged others to insert dates in their references. 



Cayley made additions to every important subject that lies within 

 the range of pure mathematics. Their importance and their amount 

 have varied in different subjects ; thus on analytical geometry his 

 writings have a dominating influence : while on the general theory of 

 functions, though he knew the subject well, he has left little mark, 

 for he concerned himself chiefly with details such as the solution of 

 more or less special problems in conformal representation. His 

 papers in general have such value that he is the author most 

 frequently quoted by the great body of current mathematicians. A 

 full record of what he has done in pure mathematics could be made 

 only by writing its history during the last half century ; all that 

 is here attempted consists of some brief indications of a selection 

 among his more obviously important contributions to mathematical 

 knowledge. 



One of the subjects with which Cayley's name will probably be 

 most closely associated is the theory of invariance. It is easy to cite 

 simple cases of what is implied by an invariantive function : two will 

 suffice. 



It is known that, in solving an ordinary algebraical equation with 

 literal coefficients, a certain functional combination of these coefficients 

 (called the discriminant) must vanish in order that two roots of the 

 equation may be equal ; for example, the equation ax~ + 2bx + c = 0, 

 has equal roots if (and only if) the quantity ac 6 2 vanishes. When 

 the variable is transformed from a* to y by a relation (l'x + m')y = 

 Ix + m, where Z, m, Z', m' are constants, then evidently two values of y, 

 corresponding to the two equal values of x, are equal. When x is 

 eliminated from the equation by means of the assumed relation, a new 

 quadratic arises having y for its variable; let it be a'y~ + 2b'y + c' = 0, 



* It was published four months after his death j only the earlier sheets had the 

 benefit of his revision. 



