196 



NA TURE 



[January 2, 1902 



the addition of granaries for dried seeds — the different 

 sorts kept separately — and with more blind passages. In 

 place of the leaves with which a mole fills its more 

 roughly constructed inner sanctuary, the kangaroo rat's 

 nest was lined with "a thick felting of fine grass and 

 weed silk, and, inside all, a lining of softest feathers." 

 "I think," he writes, "that every gay little bird on the 

 plains must have contributed one of its finest feathers to 

 that nest." 



Among the best passages in the book are those n 

 which Mr. Thompson in his first chapter reads the 

 records of the old ram's long life in the gravings of his 

 horns. The deep dent tells of the early battle in which 

 he won his spurs. The two dark-coloured, wrinkled rings 

 close together lower down are reminders of the years 

 of starvation and the sickness which carried off the 

 weaker members of the flock, and the bolder ridges wide 

 apart recall the prosperous years that followed. 



He has much to tell that is worth learning, and, left to 

 himself, can tell it e.xcellently. It will be a misfortune to 

 many lovers of natural history besides himself if Mr. 

 Thompson is beguiled into sacrificing himself on the 

 shrine of the admirers who, as he tells in his preface, 

 "bitterly denounced" him for confessing that in iin- 

 regenerate days he was not above killing a dangerous 

 wolf when he could. T. D. P. 



CUBIC AND QUARTIC CURVES. 

 An Elementary Treatise on Cubic and Quart ic Curves. 

 By A. B. Basset, F.R.S. Pp. xvi + 255^ (Cambridge: 

 Deighton, Bell and Co., 1901.) Price los. 6d. 



NOW that Salmon's "Higher Plane Curves '' is out of 

 print there is undoubtedly room for a good book 

 on the subject. The purpose of such a book would be to 

 give students who had read conic sections and the infini- 

 tesimal calculus a good knowledge of the main lines on 

 which the theory of curves has been developed. The 

 bookwork would contain discussions of the chief theorems; 

 those of less importance would be given as examples, 

 and would furnish the student with abundant matter for 

 independent thought. The proofs given would, so far as 

 possible, be models of rigour and elegance, and in the 

 rare cases where rigour was sacrificed for the sake of 

 simplicity this would be confessed. The book before us 

 has not been written altogether on these lines. There 

 are no examples, and a great deal of space is taken up in 

 proofs of the properties stated that could, in our opinion, 

 have been put to better use ; moreover, the proofs given 

 are not always satisfactory, and even the theorems them- 

 selves are sometimes wrongly stated. 



After two introductory chapters, chapter iii. deals with 

 tangential coordinates, reciprocal polars and foci, 

 chapter iv. with PUicker's equations. Then we have a 

 chapter on " cubic curves " (pp. 56-73) and another on 

 "special cubics" (pp. 74-96). The special curves dis- 

 cussed are circular cubics, and in particular some that 

 are the inverses of conies, the semicubical and cubical 

 parabolas, the folium of Descartes, the witch of .\gnesi. 

 Chapters viii., ix., x. are respectively on " quartic curves " 

 (pp. 101-132), "bicircular quartics" (pp. 133-161), 

 "special quartics" (pp. 162-204). Non-singular, or, as 

 the author prefers to call them, anautotomic quartics, 

 NO. 1679, VOL. 65] 



receive attention for three pages only (115, 117, 122). 

 The special quartics discussed are the cassinian, the 

 lemniscates of Bernoulli and Gerono, cartesians, lima- 

 (;ons, the cardioid and the conchoid of Nicomedes. 

 Chapter xi. treats of " miscellaneous curves,'' roulettes, 

 the evolute of an ellipse, the involute of a circle, the 

 catenary, tractory, elastica and spirals. Chapter xii. is 

 on projection. .Some useful references are given in 

 footnotes. 



The author has not found space for any general dis- 

 cussion of the forms of cubic and quartic curves, or of 

 the expressions for the coordinates of a variable point on 

 a curve in terms of a parameter, even when the curve is 

 unicursal. The theory of residuation is not mentioned. 

 The following are some of the matters of detail in which 

 the book might be improved. 



It is a good thing to "give special prominence to 

 geometrical methods," but we do not think it is sound to 

 estimate, say, the number of tangents that can be drawn 

 from a cusp, real or imaginary, by inspection of the 

 figure (p. 18), especially when no discussion of the form 

 of a curve near a real cusp has been gi\en ; the question 

 is in its essence an algebraical one and cannot really be 

 decided except on algebraical grounds. 



A process is given (§ 2) for finding the eliminant of 

 two binary quantics of degree n. The result would be of 

 the degree 2" in the coefficients. 



The condition given on p. 4 for the equality of r roots 

 of an equation would lead us to conclude that the 

 equation 3.r* - 4.r'' +1=0 has three equal roots. 



In the proof of Pliicker's equations (chap, iv.) it is 

 only shown that in cannot exceed w(«- i)- 28-3*, and 

 that 1 cannot exceed yi{n-2)-bh-?,K. It is not proved 

 that the curve and the Hessian meet only at multiple 

 points and points of inflexion. 



On p. 62 it is proved that the node of a nodal cubic is 

 a pole of the line of inflexions. The author must have 

 forgotten for the moment that the node is a pole of any 

 line whatever in the plane. 



Cayley's theory of conjugate poles on the Hessian of a 

 cubic is treated by means of tnlinear coordinates ; the 

 figure on p. 70 does not altogether correspond with the 

 text, for it is proved that K lies on the line PQ and that 

 C(.AMBK) is a harmonic pencil. .Also it is surely wrong 

 to say that when " .\ is given, there are in general three 

 conjugate poles corresponding to A" (p. 71). 



The proof (p. 115) that a quartic cannot have more 

 than eight real points of inflexion is very flimsy ; it con- 

 sists of an appeal to an extreme limiting case. 



The assumption that a ternary quartic can be put in 

 the form /U- + m\'- + «W- is justified by counting the 

 constants (p. 1 17), although later (p. 240) the reader is 

 very rightly warned " that counting the constants is not 

 always a safe process." 



On p. 122 we have the theorem : — 

 ".A conic can be drawn through the eight points of 

 contact of any four double tangents to a quartic." It 

 is well known that this is not true, and it is, in fact 

 inconsistent with the theorem at the foot of the same 

 page. 



There are some other points on which we do not agree 

 with the author, but notwithstanding its drawbacks, the 

 book contains much that is interesting and important. 



