8o 



NA TURE 



[November 27, 1902 



LETTERS TO THE EDITOR. 



1 The Editor does not hold himself responsible for opinions ei • 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications.] 



Classification of Quartic Curves. 



The best method of classifying curves is to commence with one 

 which is founded on properties which are unaltered by projection. 

 We thus obtain ten principal species of quartic curves, viz. 

 anautotomic, uninodal, unicuspidal, binodal, nodocuspidal, bi- 

 cuspidal. trinodal, binodocuspidal, nodobicuspidal 'and tricuspidal\ 

 but each of these species admits of a variety of subsidiary 

 divisions, owing to the fact that all curves of a higher degree 

 than the third may possess compound singularities. 



Anautotomic, unicuspidal, bicuspidal and tricuspidal quartics 

 admit of a subsidiary division depending on the number of 

 points of undulation they possess ; and it must be borne in mind 

 that, although it is convenient to use the term point of undula- 

 tion, it is the tangent at this point and not the point itself which 

 is the actual singularity. 



Uninodal quartics admit of three primary subdivisions, 

 according as the double point is an ordinary node, a flecnode 

 or a biflecnode. 



Binodal quartics admit of seven primary subdivisions, six of 

 which depend on the character of the node, whilst the seventh 

 arises from the fact that the two nodes may unite into a 

 tacnode. 



Nodocuspidal quartics admit of only four primary subdivisions, 

 three of which depend on the character of the node, whilst the 

 fourth arises from the fact that the node and cusp may unite 

 into a rhamphoid cusp. 



Trinodal quartics admit of ten primary subdivisions, and in 

 order to particularise them, we shall denote the different singu- 

 larities which involve a double point by their initial letters, 

 except that // and tc will be used to denote a triple point and a 

 tacnode cusp respectively ; so that the nomenclature ;/, n, n 

 and n, n,J will indicate that the quartic has three nodes or two 

 nodes and a flecnode respectively. We shall then have the 

 following ten species: — (i) n, n, n : (2) >i, n,f; (3) 11, n,b; 

 (4) ",/,/; (S)t>, 6, b; (6)1, u; (7) t, /; (8) /, b ■ (9) ; (10) 

 tp of the first kind. 



Binodocuspidal quartics admit of eight primary subdivisions, 

 which are as follows : — (1) c, u, n ; (2) c, n, f; (3) c, f f; (4) 

 t, c ; (5) r, n ; (6) r. f; (7) tc ; (8) // of the second kind." 



Nodobicuspidal quartics admit of three primary subdivisions, 

 which are :— (I) c, c. u ; (2) c. r ; (3) // of the third kind 



Whenever any of these primary species represents a curve 

 which has two or more points of inflection, a further sub- 

 division may usually be made which depends upon the number 

 of points of undulation it can possess. Thus the species 

 n, n, u may possess two, one or no points of undulation ; 

 whilst the species c, c, n may possess one or no such points. 



A fourth subdivision may sometimes be made which depends 

 upon whether the quartic is capable of being projected into a 

 curve which is symmetrical or hemisymmetrical with respect to 

 a pair of rectangular axes. In some cases, the possibility of the 

 projection involves the existence of compound singularities, and 

 thus the curve belongs to one of the species already considered ; 

 but in other cases, the necessary conditions do not affect the 

 singularities. Thus all trinodal quartics which are capable of 

 projection into symmetrical curves must belong to the species 

 n, n, b ; b, b, b ; or /, b, in which three respective cases the 

 quartic can be projected into the inverse of an ellipse or 

 hyperbola with respect to its centre, the lemniscate of Bernoulli 

 or the lemniscate of Gerono. On the other hand, the possibility 

 of projecting any quartic with three double points into a hemi- 

 symmetrical curve depends upon whether it can be projected 

 into the inverse of a conic with respect to a point in its axis. 

 The conditions for this do not necessarily involve compound 

 singularities, since these will only exist for special positions of 

 the centre of inversion. 



There is no necessity to adopt a classification founded on the 

 nature of the branches at infinity, since all the results can be 

 obtained by projection. Thus, if a straight line cutting in four 

 real points any quartic, which is unipartite and perigraphic, be 

 projected to infinity, the projection will be quadripartite and 

 will have four real asymptotes ; and by taking special positions 

 for the line to be projected, a variety of special results can be 



NO. I726, VOL. 67] 



obtained. By projecting a triple point or a pair of crunodes to 

 infinity, it is at once seen that a quartic can have three parallel 

 or two pairs of parallel asymptotes. Also, if the polar cubic of 

 a point breaks up in to a conic and a line cutting the quartic 

 in four ordinary points and the line be projected to infinity, 

 the projection will have four asymptotes meeting in a point. 



A quartic having three acnodes is the limiting form of an 

 anautotomic quartic in which the acnodes are replaced by three 

 perigraphic curves ; and if a line cutting the fourth portion in 

 four real points be projected to infinity, the projection will 

 be septipartite. From this it appears that the partivity of a 

 curve of the »th degree cannot be less than « + J(« - 1 ) (« - 2). 



A. B. Basset. 



Fledborough Hall, Holyport, Berks, November 14. 



The Conservation of Mass. 



Apropos of the recent discussion on the conservation of 

 mass at the Belfast meeting of the British Association, the 

 following calculation may be of interest ; it relates to the loss 

 of weight undergone by a body when raised vertically. 



Kg is the acceleration of gravity at a specified point on the 

 surface of the Earth, m the mass of a body of weight w, then 



w = mg. 



Now let the centre of gravity of the body be raised through a 

 vertical distance d ; g will be changed into 



Ar+J- 



R being the radius of the Earth (supposed spherical), and the 

 corresponding weight of the body will be 



on the supposition of the conservation of rn u 

 The loss of weight is thus 



S = w-w' = wi 1- I 



I (R + ^) a / 



2dw 



"{-Krp-f 



neglecting second and higher powers of — . 



R 

 As a particular example, take w _= 1 kilogm. , d= 10 cm. and R 

 approx. =6357 x 10 s cm. 

 Then 



5 = o'oooo3 gm. 



[The term involvi 



"g (ff w 



•ould have the first significant 



figure in the fifteenth place, and therefore we were justified 

 in this case in neglecting it.] 



This small difference is, I believe, of the same order as those 

 which Prof. Landolt found ; but the ratio of the difference to 

 the whole weight (i.e. 2d: R) must have been much greater in 

 his experiments. Although Prof. Lindolt's discrepancies miy 

 receive a perfectly different explanation, it is quite conceivable 

 that a balance could be constructed which would detect such 

 small differences. It is scarcely necessary to point out that, in 

 the actual performance of the experiment, the scale-pin contain- 

 ing the counterpoising weights must be at the same height 

 during the two weighings. D. M. Y. Sommerville. 



St. Andrews, November 12. 



A Simple Experiment in Diffraction. 



M. G. Fousserau describes, in the Journal dc Physique for 

 October, a simple apparatus for viewing diffraction and inter- 

 ference phenomena, a modified form of which I have experi- 

 mented on with success. In the latter form, the source of light 

 was obtained by placing a diaphragm on the stage of a micro- 

 scope, on which sunlight was concentrated by means of the 

 mirror and condenser, and the diffraction effects were produced 

 by placing perforated pieces of tinfoil on the top of the micro- 

 scope tube where the eye-piece is usually placed. On placing 

 the eye close up to the tiny hole in the tinfoil, various diffraction 

 patterns were seen. The difficulty of piercing a hole that is truly 

 circular in tinfoil made it hard to obtain perfect rings, but the 

 " failures " were often very interesting. A rectangular aperture 



