January 13, 1899.] 



SCIENCE. 



59 



Laboratory Notes,' A. C. LaEgmuir ; 

 ' Flame Colorations by Bromides and 

 Chlorides of Nickel and Cobalt,' A. S. 

 Cushman ; ' Classen's Reaction as an 

 Aid to Determination of Constitution of 

 Terpene Ketones,' M. C. Burt; ' Sixth An- 

 nual Report of Committee on Atomic 

 Weight,' F. W. Clarke. 



A luncheon was provided by the New 

 York Section, wliich was served in the 

 Industrial Laboratory, after which visits 

 to various manufacturing establishments 

 and a demonstration of the properties of 

 liquid air at the College of the City of 

 New York occupied the rest of the day, 

 and a dinner at the Waldorf-Astoria in 

 the evening closed the official program of a 

 meeting which had been successful beyond 

 the expectations of the most sanguine of 

 those who had woi-ked for it. 



The attendance was not less than one 



hundred and fifty at any of the sessions, 



and among them a number of ladies, who 



also graced the dinner with their presence. 



DuRAND Woodman. 



Secretary 



SCIENTIFIC BOOKS. 

 The Collected Mathematical Papers of Arthur 

 Cayley. 4to. 13 Vols., each $6.25. Supple- 

 mentary Vol., containing Titles of Papers 

 andlndex. New York, Macmillan Co. $2.50. 

 This republication by the Cambridge Univer- 

 sity Press of Cayley 's papers, in collected form, 

 is the most fitting monument of his splendid 

 fame. 



He must ever rank as one of the greatest 

 mathematicians of all time. Cayley exceedingly 

 appreciated this action of the Syndics of the 

 Press, and seven of the large quarto volumes 

 appeared under his own editorship. 



As to what these thirteen volumes contain it 

 seems vain to attempt even a summary. They 

 cover the whole range of pure mathematics, 

 algebra, analysis, mathematical astronomy, dy- 

 namics, and in particular groups, quadratic 

 forms, quan tics, etc. , etc. 



Though abreast of Sylve,ster as an analyst, he 



was, what Sylvester was not, also a geometer. 

 Again and again we find the pure geometric 

 methods of Poncelet and Chasles, though, per- 

 haps, not full assimilation of that greater one 

 than they who has now absorbed them — von 

 Staudt. 



Cayley not only made additions to every im- 

 portant subject of pure mathematics, but whole 

 new subjects, now of the most importance, owe 

 their existence to him. It is said that he is 

 actually now the author most frequently quoted 

 in the living world of mathematicians. His 

 name is, perhaps, most closely linked with the 

 word invariant, due to his great brother- in- arms, 

 Sylvester. 



Boole, in 1841, had shown the invariance of 

 all discriminants and given a method of deduc- 

 ing some other such functions. This paper of 

 Boole's suggested to Cayley the more general 

 question, to find ' all the derivatives of any 

 number of functions which have the property of 

 preserving their form unaltered after any linear 

 transformation of the variables.' His first re- 

 sults, relating to what we now call invariants, 

 he published in 1845. A second set of results, 

 relating to what Sylvester called covariants, he 

 published in 1846. Not until four or five years 

 later did Sylvester take up this matter, but 

 then came such a burst of genius that after his 

 series of publications, in 1851-4, the giant theory 

 of Invariants and Covariants was in the world 

 completely equipped. 



The check came when Cayley, in his second 

 Memoir on Quantics, came to the erroneous 

 conclusion that the number of the asyzygetic 

 invariants of binary quantics beyond the sixth 

 order was infinite, ' thereby,' as Sylvester says, 

 'arresting for many years the progress of the 

 triumphal car wliich he had played a principal 

 part in setting in motion.' 



The passages supposed to prove this are 

 marked ' incorrect ' in the Collected Mathemat- 

 ical Papers. But this error was not corrected 

 until 1869 [Crelle, Vol. 69, pp. 323-354] by 

 Gordan in his Memoir [dated 8th June, 1868] : 

 " Beweis dass jede Covariante und Invariaute 

 einer binaeren Form eine ganze Function mit 

 numerischen Coefiicienten einer endbchen Au- 

 zabl solcber Formen ist." 



Cayley at once returned to the question , found 



