Septembek 16, 1904.] 



SCIENCE. 



365 



richer in curves of the second degree. The 

 curves with real center are the ellipse (real, 

 semi-real, ideal or imaginary) with the 

 important varieties, circle and hypercycle 

 (^equidistant from the straight). The 

 curves denuded of center, even at infinity 

 are the parabolas (elliptic, veritable, hy- 

 perbolic). The curves with center situated 

 at infinity are the oriconics (oriellipse, with 

 the variety orieycle of Lobaehevski, orihy- 

 perbola). 



"M. Barbarin generates the lines of the 

 second degree by homography and by 

 movements of linkages; he investigates 

 their foci, their focal circles and their di- 

 rectrices; finally their reciprocal curves. 

 Two reciprocal curves of the second de- 

 gree are such that each is the locus of the 

 center (real or ideal) of the tangents of 

 the other. The properties of these curves 

 are a consequence of the principle of 

 duality, which is, so to say, evident in non- 

 Euclidean geometry. 



"The author then studies the plane sec- 

 tions of a cylinder or of a cone of the 

 second degree (that is to say, having for 

 plane directrix a curve of the second de- 

 gree) ; he finds again in this way all the 

 varieties of curves of the second degree, 

 which are, therefore, truly conies. He 

 extends to non-Euclidean conies the most 

 celebrated theorems relative to Euclidean 

 conies and, in particular, those of Dandelin. 

 He obtains in a manner more systematic 

 still all the curves of the second degree, 

 Riemannean, Euclidean, Lobachevskian, by 

 cutting the cone of the second degree by a' 

 concentric sphere, the common center being 

 real, at finite or infinite distance, or ideal. 



"One again finds conies in cutting by a 

 plane the straight equidistant surface (tube 

 of revolution with rectilinear axis, or hyper- 

 cyeloide of revolution). In Riemannean 

 geometry there is a case where one finds as 

 section two straights equidistant from the 

 » axis, but not eoplanar with the axis (com- 



pare above, II., 5) : these straights are the 

 helices of this surface. 



"B. Quadrics. — The reduction of the 

 general equation of the surfaces of the 

 second degree is deduced from the discus- 

 sion of the equation in s of the nth degree. 



' ' In Riemannean space, we find two prin- 

 cipal species — ellipsoid (with the varieties 

 ellipsoid of revolution, tube sphere), pipe- 

 hyperboloid (with the varieties cone, hy- 

 perboloid of revolution or elliptic tube, two 

 planes). 



"In Lobachevskian space we find first 

 the species ellipsoid (with three unequal 

 real axes, semi-real with two real axes, 

 with one real axis or imaginary) ; the first 

 hyperioloid (with one nappe real, with two 

 nappes real, with one nappe ideal) ; the 

 second hyperboloid (with two nappes real 

 or ideal) . The varieties or limits of these 

 three species are very numerous. All these 

 surfaces have a center and three principal 

 planes. 



"The paraboloids (elliptic, semi-elliptic 

 or hyperbolic) and their numerous varieties 

 have no center and have two principal 

 planes. They cm< the sphere of infinite 

 radius. / 



' ' At the limit, when they become tangent 

 to this sphere, they are transformed into 

 oriquadrics (oriellipsoid, orihyperboloid) 

 and into their varieties. 



"M. Barbarin has studied the rectilinear 

 and circular sections of these surfaces, their 

 focal spheres and their directrices. 



"IV. Infinitesimal geometry. 1. Meas- 

 ure of areas and of volumes (List No. 4, 

 pp. 164-167; No. 6; No. 7, pp. 50-59).— 

 M. Barbarin in his little book, 'The non- 

 Euclidean Geometry,' summarizes in some 

 pages the results found by Lambert, Loba- 

 ehevski, Simon, etc. ; but he calls attention 

 also (List No. 6) to an original idea of 

 which he is the author. He has remarked 

 that the volume of a frustum of a non- 

 Euclidean cone of revolution is propor- 



