NOTES AND REFERENCES. 
1. As to the history of Determinants, see Dr Muir’s “ List of Writings on 
Determinants,” Quart. Math. Jour. vol. xvn. (1882), pp. 110—149; and the interesting 
analyses of the earlier papers in course of publication by him in the R. 8. E. Proceedings, 
vol. xiii. (1885—86) et seq. 
The (new ?) theorem for the multiplication of two determinants was given by 
Binet in his “ Mémoire sur un système de formules analytiques &c.” Jour. École 
Polyt. t. x. (1815), pp. 29—112. 
An expression for the relation between the distances of five points in space, 
but not by means of a determinant or in a developed form, is given by Lagrange 
in the Memoir “ Solutions analytiques de quelques problèmes sur les pyramides 
triangulaires,” Mém. de Berlin, 1773: the question was afterwards considered by 
Carnot in his work “ Sur la relation qui existe entre les distances respectives de 
cinq points quelconques pris dans l’espace, suivi d’un essai sur la théorie des trans 
versales,” 4to Paris, 1806. Carnot projected four of the points on a spherical surface 
having for its centre the fifth point, and then, from the relation connecting the 
cosines of the sides and diagonals of the spherical quadrilateral, deduced the relation 
between the distances of the five points : this is given in a completely developed 
form, containing of course a large number of terms. 
Connected with the question we have the theorem given by Staudt in the paper 
“Ueber die Inhalte der Polygone und Polyeder,” Grelle t. xxiv. (1842), pp. 252—256; 
the product of the volumes of two polyhedra is expressible as a rational and 
integral function of the distances of the vertices of the one from those of the other 
polyhedron. 
More general determinant-formulæ relating to the “powers” of circles and spheres 
have been subsequently obtained by Darboux, Clifford and Lachlan : see in particular 
Lachlan’s Memoir, “ On Systems of Circles and Spheres,” Phil. Trans, vol. clxxvii. 
(1886), pp. 481—625. 
2 and 3. The investigation was suggested to me by a passage in the Mécanique 
Analytique, Ed. 2 (1811), t. I. p. 113 (Ed. 3, p. 106) ; after referring to a formula 
of Laplace, whereby it appeared that the attraction of an ellipsoid on an exterior 
point depends only on the quantities B 2 — A 2 and C 2 — A 2 which are the squares of