532 TRANSACTIONS OF SECTION A. 



The following table gives, for comparison, the number of titles in these 

 bibliographies, and, in brackets, the number which the present writer has collected 

 within the corresponding limits of time and subject-matter : — 



Th of P. and NDim 



N.-L<. Geom. 

 StackelandEngel . . . .273 (330) 



Halsted 112 (400) 76 (130) 



Schlegel — 439 (860) 



Bonola 915 (1250) 



The present writer's collection, which is made up to 1907 (the last year of the 

 ' Jahrbuch '), contains about 3,500 titles, composed roughly as follows : Theory of 

 parallels, 600; non-euclidean geometry and foundations of geometry, 1,300; 

 n dimensions, 1,600. The arrangement of the material is in three parts, as in 

 Riccardi's Euclidean Bibliography. 



1. A chronological catalogue, the titles in each year being arranged alphabeti- 

 cally under the authors. 



2. An author-index. This contains the full names of the authors with dates 

 of birth and death, and the abbreviated titles with their dates. This index could 

 be greatly condensed by giving only the dates of the papers, but a comparison of 

 the ' Namenregister ' of the 'Jahrbuch' with the 'Liste des Auteurs ' of the 

 'Revue Semestrielle ' shows the great advantage of the former system. 



3. A subject-index. The classification used is that of the 'Index du Reper- 

 toire Bfbliographique,' the new edition of which in 1908 was specially extended to 

 treat adequately of hyperspace. The references here are by the author's name, 

 the date and the number as ' Mansion 1S96V The class letters of this system 

 are affixed to the titles in the chronological catalogue. It has been found 

 advisable to classify somewhat more minutely than according to the headings of 

 the 'Index ' — for example, Qlb (hyperbolic geometry) is sub-divided according to 

 the divisions of K l (elementary geometry), L' (conies), &c. 



The foregoing considerations will show the scope, if not the demand, that 

 exists for such a bibliography. The arguments for a new bibliography may be 

 shortly summed up : — 



1. The increasing importance and wide application of this branch of mathe- 

 matics. 



2. The convenience of having all the literature collected in one place, thus 

 avoiding a prolonged search through a long series of volumes, and giving the 

 literature of all time, whereas the periodicals, 'Jahrbuch,' &c, have a definite 

 beginning. 



3. The necessity for a subject-index (which none of the existing bibliographies 

 possess), this being the most important and useful, as well as the most difficult, 

 portion of the bibliography. 



8. Report on the History and Present State of the Theory of Integral 

 Equations. By H. Bateman. — See Reports, p. 345. 



9. The Foci of a Circle in Space and some Geometrical Theorems 

 connected therewith. By H. Bateman. 



In a space of three dimensions the centres of the two spheres of zero radius 

 (or isotropic cones) which pass through a circle have been called by Darboux 

 the foci of the circle. Many focal properties of curves and surfaces may be 

 studied with the aid of the foci of systems of circles — for instance, the foci of the 

 circular sections of a conicoid lie on the focal conies, and conversely the circles 

 whose foci are the extremities of a system of parallel chords of a conic generate 

 a conicoid, and by varying the direction of the chords a confocal system of 

 conicoids is obtained. 



In a space of four dimensions the foci of a sphere may be defined to be 

 the centres of the two hyperspheres of zero radius which pass through the 



