Notices respeclmg New Books, 297 



a metric process, a system of txm reciprocal ellipsoids, derived 

 from one Jixed sphere, and of determining, also graphically, 

 for each point on either ellipsoid, the reciprocal point on the 

 other. 



Inscribe in the fixed sphere a plane quadrilateral (LyMyL/M/), 

 of which the four sides (l^M;, m^l/, l/m/, m/l^) shall be re- 

 spectively parallel to four fixed right lines (ab, ac', ab', ac), 

 diverging from the centre (a) of the sphere; and prolong 

 (if necessary) the first and thirtl sides of this inscribed qua- 

 drilateral, till they meet in a point e ; and the second and 

 fourth sides of the same quadrilateral, till they intersect in 

 another point h. Then these two points, of iniersection k and 

 H, thus found from two 2>airs of opposite sides of this inscribed 

 quadrilateral, will be two reciprocal points on two reciprocal 

 elli-psoids', which ellipsoids will have a common mean axis, 

 namely that diameter of the fixed sphere which is perpendi- 

 cular to the plane of the four fixed lines : and those lines, ab, 

 ac', ab', AC, will be related to the two ellipsoids which are 

 thus the loci of the two points e and h, according to the laws 

 enunciated in article 61, in connexion with a different con- 

 struction of a system of two reciprocal ellipsoids (derived there 

 from one common moving sphere) ; which former construction 

 also was obtained by the aid of the calculus of quaternions. 

 Thus the lines ac, ac' will be the two cyclic normals of the 

 ellipsoid which is the locus of e, but will be the axes of cir- 

 cumscribed cylinders of revolution, for that reciprocal ellipsoid 

 which is the locus of h; and conversely, the lines ab, ab' will 

 be the axes of the two cylinders of revolution circumscribed 

 about the ellipsoid (e), but will be the cyclic normals, or the 

 perpendiculars to the cyclic planes, for the reciprocal ellip- 

 soid (h). 



[To be continued.] 



XLIII. Notices respecting New Books. 



Letters addressed to H.R.H. the Grand Duke of Saxe Coburg and 

 Gotha, on the Theory of Probabilities, as applied to the Moral and 

 Political Sciences. By M. A. Quetelet, Astronomer Royal of 

 Belgium, Corresponding Member of the Institute of France, 8(C. SfC. 

 Translated from the French by Olinthus Gregory Downes, of the 

 Economic Life Assurance Society. 



OF this work, which was begun by M. Quetelet in 1837, and 

 published at Brussels, we believe, early in 1845, the author 

 thus describes the object in his preface. 



" Certain circumstances, which have left me many pleasant remi- 

 niscences, made it necessary for me nearly ten years since to devote 



