384 Notices respecting Neio Books. 



point. This species may be considered as formed by the planes 

 of two tetrahedra having a common base. 



(b) If these three lines intersect in two points, but do not lie 

 in the same plane, there will in general be four diagonal planes 

 only, and the four linear diagonals will intersect in four points 

 not in the same plane. 



(c) If only two of the three lines intersect, there will in 

 general be only two diagonal planes, upon each of which one 

 pair only of diagonal lines will intersect. 



(d) Lastly, if none of the three lines meet, there will in 

 general be no diagonal planes, and the four diagonal lines will 

 not meet. 



Each of these four species may, however, have singular cases. 



8. In species (a) all parallel or subcontrary planes, which 

 divide or cleave the hexahedron into two other hexahedra, have 

 similar quadrilaterals traced upon them. In the other three 

 species this is only true, in general, of selected planes. In the 

 singular case of the inscribed tetrahedron, the quadrilaterals ob- 

 tained by parallel cleavage are not always similar. 



9. The hexahedron inscribed in the sphere belongs in general, 

 as has been seen, to species (a). Its two tetrahedra have the 

 corresponding plane angles at their vertices supplementary each 

 to those of the other. The singular case is a very restricted 

 singular form of sj)ecies (c), or it may be looked upon as an in- 

 determinate form, arising out of singularity, in species (a) ; i. e. 

 that when two of the three associated lines are parallel, the third 

 may leave their plane, under certain conditions of symmetry. 



30 Scarsdale Villas, Kensington, "W., 

 October 18, 1861. 



L. Notices respecting New Books. 



An Elementary Treatise on the Theory of Equations, with a Collection 

 of Examples. By I. Todhunter, M.A. 



]Y|R. TODHUNTER'S merits, as a writer of some of our best 

 ■!•▼•*■ elementary treatises on mathematics, are now so well esta- 

 blished as to render it quite unnecessary to dwell upon the manner 

 in which this, his last task, has been performed. It will suffice, 

 therefore, to inform the mathematical student that a thoroughly 

 trustworthy, complete, and yet not too elaborate, treatise on the 

 Theory of Equations is now within his reach ; that, as far as the 

 elementary character of the work would permit, the treatment of the 

 subject has been brought up to the level of the science of our day; and 

 that, in some branches of the subject, the more elaborate researches 

 of modern authors have been carefully examined, their suitable por- 

 tions judiciously selected, and now for the first time collected. 

 The three chapters on Determinants will be particularly accept- 



