VI 



NA TURE 



[Supplement, 

 laniiary 18, 1S94 



tion the later work of Halphen, Neither, \'a!cntine, and 

 Hilbert. These authors make considerable use of 

 Cayley's conception of the monoid surface, the curve 

 being regarded as the partial intersection of a 

 cone with a monoid surface ; Halphen obtained the 

 fundamental theorem which shows the existence of a 

 monoid surface of order n -\- i, where « is the order of 

 the cone of lowest order which passes through all the 

 nodal lines of the aforesaid cone. Important re- 

 marks are made on the theory arising from this notion. 

 In the Professor's view, the question of the classiJicatio7i 

 of curves in space, according to the representation, as the 

 partial intersection of a cone and a monoid surface is 

 that which properly first presents itself. 



There ai e many papers on geometry, including those on 

 " skew surfaces ' and " sextactic points of a plane 

 curve," with which all geometricians are familiar. 



The paper No. 347, " On the Notion and Boundaries 

 of Algebra,'' considers in particular the line of separa- 

 tion between finite and transcendental analysis. The 

 views of so eminent a man on this subject are neces- 

 sarily of interest and importance. In stating algebra 

 to be both an ait and a science, the author is in 

 agreement with the constantly reiterated opinion of 

 Prof. Sylvester, and indeed with that of the great 

 majority of those who have made algebra a subject of 

 special study. The two great divisions of algebra are 

 stated to be ''tactic'" and "logistic," and the remark 

 is made that every algebraical theorem rests, ultimately, 

 on a tactical foundation. Mathematicians will, we think, 

 perceive in the word " tactic " a singularly felicitous 

 expression to denote those operations which relate to 

 arrangement of mat rial. 



The paper No. 312, " On the Partitions of a Close," 

 generalises Euler's polyhedral relation, 



F + S = E + 2 



and is a first step towards Listing's well-known develop- 

 ments. 



The frontispiece of vol vi. is a portrait of Prof. Cayley, 

 attired in gown, seated at his desk in the act of writing. 

 The picture is somewhat dark and not quite so pleasing ' 

 as those which of recent years have appeared in 

 Nature and in the American Journal of Mathematics. 

 The volume is principally on geometry. The memoir 

 No. 412, on "Cubic Surfaces," adopts a classification 

 depending on the nature of the singularities. In the 

 notes the notation is compared with that of Zeuthen, and 

 several apparent discrepancies in the results are 

 explained. 



There is a long memoir on the polyzomal curves 



VU + n'V + . ... =0 



U, V, &c., being rational integral functions of the same [ 

 degree in the variables. The general 2/-zomal curve {i.e. 

 T functions U, V, &c.) is considered, and the branches, 

 singularities, order, class, &c., determined. The investi- 

 gation is intimately connected with Casey's work, on 

 BicircularOuartics, published in 1S67 in the Proceedings 

 of the Royal Irish Academy. 



Other geometrical subjects considered are " Reciprocal 

 Surfaces," " Skew and Developable Surfaces," " Abstract 

 Geometry," "Cubical Divergent Parabolas," &c. In 

 NO. 1264, VOL. 49] 



analysis we have the eighth memoir on quantics, which 

 relates chiefly to the binary quintic ; and, finally, men- 

 tion may be made of the reproduction of Euler's memoir 

 of 1758, on the rotation of a solid body. 



The number of papers which appear in the six volumes 

 is 416. Several hundreds of papers have yet to appear, 

 and it seems improbable that a total of ten volumes will 

 be found sufficient. This improbability is increased from 

 the circumstance that Prof. Cayley is still producing a 

 considerable amount of mathematical work. 



For excellence of mathematical printing, and general 

 care of production, unstinted approval may be awarded 

 to the Cambridge University Press. These handsome 

 volumes, as they appear, are rearing a fitting monument 

 of the work of an eminent man, and are causing gratifi- 

 cation and congratulation amongst mathematicians the 

 world over. P. A. MacMahon. 



THE PAMIRS. 

 The Pamirs : bcin^ a Narrative of a Year's Expedition 

 on Horseback and on Foot tJirougli Kashmir, Western 

 Tibet, Chinese Tartary, and Russian Cental Asia. By 

 the Earl of Dunmore, F.R.G.S. In two volumes. 

 (London : John Murray, 1893.) 



LORD DUNMORE embodies in these volumes the 

 journal of a somewhat remarkable journey of a 

 year s duration, the initial and terminal points of which 

 were Karachi and Constantinople. During most of the 

 time he was accompanied by Major Roche, and the object 

 of the journey was sport, especially the pursuit of the 

 Ovis poli. Unfortunately this great sheep— of which 

 Lord Dunmore (vol. ii. p. 56) appears to doubt the ovine 

 character — inhabits a country the political geography of 

 which is still undergoing spasmodic evolution, and there 

 is reason to suspect the censorship of the Indian Govern- 

 ment on all English writings bearing on the region. Thus 

 we cannot expect any great contributions to the detailed 

 topography of the Pamirs to be made public, and as the 

 author makes no claim to any scientific acquirements, 

 the botany and geology of the vast tracts wandered over 

 are left no clearer than they were before. Major Roche, 

 however, made an entomological collection. It is deeply 

 to be deplored that every traveller, who is so fortunately 

 circumstanced as to be able to push his way where 

 few intejligent Europeans have been before, should 

 not make some sort of preparation to fit himself 

 for utilising his rare opportunities. For such pre- 

 liminary instruction in science the Royal Geographical 

 Society has made ample provision specially adapted 

 to the wants of travellers, and as years go on we 

 trust that the imputation of ignorance of what and how 

 to observe may no longer be the necessary prelude to 

 the criticism of works of travel in a scientific journal. 

 Major Roche was fortunately provided with a camera, 

 and made splendid use of it, although most of his negatives 

 were unfortunately destroyed. Lord Dunmore himself 

 is something of an artist, and much value must be 

 attached to the landscapes which illustrate his book. 

 Both travellers carried aneroids and, presumably, some 

 surveying instruments, as a map with several new features 

 is one of the results of the journey. We may point out 

 that the larger- scale map of the Pamirs should have been 



