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begun. If a text-book had been taken up at the beginning, with the 

 intention of following it, that text-book was most likely doomed to 

 oblivion for the rest of the term, or until the class had been made 

 listeners to every new thought and principle that had sprung from 

 the laboratory of his mind, in consequence of that first difficulty. 

 Other difficulties would soon appear, so that no text-book could last 

 more than half the term. In this way the class listened to almost all 

 of the work that subsequently appeared in the journal. It seemed to 

 be the quality of his mind that he must adhere to one subject. He 

 would think about it, talk about it to his class, and finally write 

 about it for the journal. The merest accident might start him, but, 

 once started, every moment, every thought was given to it, and, as 

 much as possible, he read what others had done in the same direc- 

 tion ; but this last seemed to be his weak point ; he could not read 

 without meeting difficulties in the way of understanding the author. 

 Thus, often his own work reproduced what others had done, and he 

 did not find it out until too late." 



Dr. W. P. Durfee, Professor of Mathematics at Hobart College, 

 Geneva, N.Y., has written : 



" His manner of lecturing was highly rhetorical and elocutionary. 

 When about to enunciate an important or remarkable statement he 

 would draw himself up till he stood on the very tips of his toes, 

 and in deep tones thunder out his sentences. He preached at us 

 at such times, and not infrequently he wound up by quoting a few 

 lines of poetry to impress upon us the importance of what he had 

 been declaring." 



On the death of H. J. S. Smith, Sylvester was elected to the 

 Savilian Professorship of Geometry at Oxford, and in December, 

 1883, he finally left Baltimore to enter upon residence in New- 

 College, Oxford. At the time his mind was occupied with the 

 theory of a new species of invariants ; these are differential and of 

 more immediate application to geometry than those of pure algebra. 

 Sophus Lie had treated the whole subject of differential invariants 

 from a general point of view, and had given the various categories^, 

 but had made no attempt to develop the special case treated by 

 Sylvester. His notice appears to have been first attracted to the 

 subject by the well-known invariantive property of the differential 

 expression known as the Schwarzian Derivative, which in this country 

 had been studied by Cayley and Forsyth. The invariantive forms, he 

 quickly reached, he termed reciprocants, the name arising from the 

 fact that, from his original point of view, the expressions arrived at 

 were unchanged, to a factor pres, by the simple interchange of the 

 dependent and independent variables. Later he considered the 

 general linear and homographic transformations applied to similar 

 forms, and propounded an extensive theory of great geometrical 



