Royal Society. 241 



for a student who should attempt this would find his progress stayed 

 at the beginning of the second chapter. This, however, does in no 

 way detract from the merit of the book, which, we repeat, must be 

 considered as a complement and a valuable step in advance. 



The terms Trilinear coordinates, Anharmonic ratio, Involution, 

 Reciprocal Polars, &c, have been for some years familiar terms in 

 the studies of the University of Cambridge ; and those who have 

 read Salmon's 'Conic Sections' or Todhunter's 'Coordinate Geometry,' 

 know the immense power they confer as a means of investigation ; 

 but we meet here with a term which to many students will be a new 

 one, although the subject owes its existence and vitality chiefly to 

 the labours of our own countrymen, Sylvester, Salmon, Boole, Spot- 

 tiswoode, Cayley, and others. What is a determinant ? Answer : 

 Write down n rows of symbols with n symbols in each row, and en- 

 close the whole between two vertical lines. That is a determinant. 

 It is a conventional form of expressing in a concise manner a com- 

 plicated function of these n symbols ; and these same functions are 

 so frequently recurring, not only in investigations concerning curves, 

 but in almost every branch of mathematical inquiry, that an abridged 

 notation for them was absolutely needed. A determinant of 5 rows, 

 and therefore containing 25 symbols, would, if written at full length, 

 contain 120 terms with 5 symbols in each term, i. e. 600 symbols 

 instead of 25. Chapter III. of the book is devoted to a clear exposi- 

 tion of the simplest laws of combination of these functions, and will 

 serve as a most useful introduction to the study of many modern 

 scientific memoirs. We are only sorry that Mr. Ferrers does not 

 dwell at greater length on them, and give us exact proofs of some 

 of the remarkable results to be -found in Spottiswoode, Salmon, 

 Brioschi, Crelle's Journal, &c. 



We cordially recommend this little work to those of our readers 

 who have mastered the ordinary coordinate geometry. 



XXXI. Proceedings of Learned Societies. 



ROYAL SOCIETY. 

 [Continued from p. 164.] 

 November 22, 1860. — Major-General Sabine, R.A., Treasurer and 

 Vice-President, in the Chair. 



r PHE following communications were read : — 

 ■* " Researches on the Phosphorus-Bases." — No. VIII. Oxide 

 of Triethylphosphine. By A. W. Hofmann, LL.D. Received 

 July 24, 1860. 



In our former experiments *, Cahours and myself had often 

 observed this substance, but we did not succeed in obtaining it in a 

 state of purity fit for analysis. Nevertheless, founding our conclu- 

 sion on the composition of the corresponding sulphur-compound, 

 and having regard to the analogies presented by the corresponding 

 terms of the arsenic- and antimony-series, we designated this body as 

 * Phil. Trans. 1857, p. 575. 



