322 



NATURE 



[May 13, 1920 



which may break up into a number of distinct, 

 irreducible curves. How far any one of these 

 curves should be considered to form a part of the 

 envelope proper depends upon our definition of 

 "envelope." For instance, in the author's 

 example (p. 61) — 



(y 



\^-X^: 



the a-eliminant is a;3 = o, which is a cusp-locus: 

 Mr. Fowler refuses to regard this as an envelope, 

 but if we regard the cusps as limiting forms of 

 nodes, we may fairly regard x = o as the limit of 

 an envelope. However, this is a matter of slight 

 importance, because each case that occurs can 

 be treated independently. 



On p. 60 we have the example — 



o2/+(2o+i)fe=o, 



where the o-eliminant is h{h-f)z=o, and neither 

 h = o nor h-f = o is an envelope. If we put 

 {2a+i)/a^ = P, the family of curves is f+fih = o, 

 and the j8-eliminant is h = o. It seems worth 

 while to direct attention to this apparent dis- 

 crepancy, because similar cases occur in problems 

 of maxima and minima, etc. If, starting with 

 f+$h = o, we replace )8 by (20+ i)/a2, we obtain, 

 by variation of a, the same pencil of curves ; but, 

 generally speaking, each curve occurs twice, and, 

 as a rule, for different values of o. U $=-i, 

 (a +1)2 = 0, and the curve f-h = o counts twice 

 for the double value a=-i, and hence f-h = o 

 occurs in the o-eliminant, though it does 

 not appear in the )8-eliminant. Similar results, of 

 a more complicated kind, occur ii in f+ $hz=o we 

 put $z=<f>{a)/il/{a), where cf>{a), xj/{a) are any poly- 

 nomials in a. 



In his last two chapters we think Mr. Fowler 

 has rather lost his sense of proportion. In the 

 eight pages devoted to the singular points of 

 plane curves, scarcely anything more is attempted ! 

 than a discussion of ordinary nodes and cusps; ' 

 on the other hand, fourteen pages are filled with 1 

 the theory of rectilinear and curvilinear asymptotes, | 

 and many of the results may fairly be said to [ 

 be more interesting in function-theory than in i 

 geometry proper. ' 



The author has conscientiously given references I 

 to the text-books which he has more or less fol- 

 lowed in his exposition; but there is no biblio- 

 graphy of original papers, such as add greatly 

 to the value of other tracts in this series. We 

 hope that in future editions this want will be 

 supplied; reference should at least be made to 

 Puiseux, Weierstrass, Smith, and Halphen in con- 

 nection with singular points. G. B. M. 

 NO. 2637, VOL. 105] 



A Garden in the Dunes. 



Arcachon^ Ville de SanU: Monographic Scien- 

 tifique et Medicate. By Dr. F. Lalesque. 

 Pp. viii + 798. (Paris: Masson et Cie, 1919.) 

 Price 25 francs net. 



THIS handsome volume is mainly a justification 

 of Arcachon as a health-resort, and it should 

 appeal to those who feel the attractions of a life 

 in France, and yet shrink from the crowd and 

 cosmopolitan gaiety of the Mediterranean Riviera. 

 The author is not content, however, with giving 

 climatic temperature-charts, records of rainfall,, 

 and views of exercises on the shore and of yachting 

 on the broad sea-basin. He has made a "regional 

 survey " of his district, and the details of the 

 natural history will provide matter of much 

 interest to those who sojourn in the town. The 

 modern " town " is a veritable garden city,, 

 especially developed in the last twenty years 

 in accordance with Dr. Lalesque 's scheme for la 

 cure lihre, and the separate villas provided in the 

 ville d'hiver among the woods offer the patient the 

 cheerful encouragements of family life in place 

 of the more formal control of the sanatorium. 

 The sand-dunes have been captured, as it were, 

 and converted into shelters, while the fishing 

 village on the open northern strand has been 

 enlarged as a place for summer bathing. 



Arcachon, indeed, began its career of usefulness 

 when Francois Legallais, a retired sea-captain, 

 imported from India to its desolate sand-hills the 

 idea of the bungalow in 1823. Dr. Lalesque gives 

 us, in a few brief lines, a sketch of this rather 

 captivating incident in human settlement and 

 geography. His range of vision is wide. He 

 interests us equally in the wind-worn grains of 

 magnetite brought northward from the great fan- 

 deltas of the Adour system; in the "alios," a 

 ferruginous conglomerate formed as an iron-pan 

 in the subsoil, the permeability of which has now 

 been triumphantly established; in the health of 

 the oyster, an inoffensive creature, infected with 

 typhoid germs entirely by the carelessness of man ; 

 and in the diatoms that flourish in the Lac de 

 Cazeaux, to the actual benefit, it appears, of the 

 water-supply of Arcachon. Dr. Lalesque in 1890 

 made an independent investigation of the irritating 

 power on the human skin of the processionary 

 caterpillar, the larva of Bombyx pityocampa, 

 which inhabits the pine-trees of the coast, and he 

 concludes that the hairs which cause urticaria are 

 scattered from the nests of the insect by the wind. 

 Even this affection seems trivial at Arcachon ; we 

 can imagine a visitor, temporarily inclined to 

 Irritation, being calmed by Dr. Lalesque 's 



