Notices respecting JSew Books. 431 



distinguished from other textbooks on the same subject by con- 

 taining an exposition of Gibbs' Vector Method, so the third edition 

 offers a special attraction in the lecture (No. 82) dealing with 

 Leuschner's method, and an appendix of 69 pages containing a 

 collection of the formulae to be applied in different cases, and 

 a series of seven numerical examples fully worked out in detail. 

 Of the last two methods — Harzer's and Leuschner's — the late 

 M. Poincare has said " MM. Harzer et Leuschner paraissent avoir 

 realise un progres notable sur les methodes usuelles "; and of those 

 who are familiar with Prof. Harzer's contribution to the Astronom- 

 ische Nachrichten, No. 3371, " Ueber eine allgemeine Methode der 

 Bahnbestimmung," some may be inclined to regret that room was 

 not found in this volume for a lecture on his method as well as 

 that of the American astronomer. But, indeed, it might well 

 appear unreasonable to complain of omissions in a work containing 

 so much that is new and valuable, and the writer explains at 

 length in his introduction the reasons which guided him in the 

 selection of his material. 



The objections to be urged against Harzer's method are chiefly 

 of a practical kind. It presupposes in the most advantageous case 

 five complete observations of the planet, and the amount of labour 

 entailed in the computations is large in proportion to the increased 

 accuracy to be gained. It seemed, however, to Leuschner that, 

 if freed from its practical disadvantages, Harzer's method would 

 permit of the determination of an orbit, without previous hypo- 

 thesis as to the eccentricity, more readily even than Olber's classical 

 method for determining parabolic elements in the case of comets. 



The improvements aimed at by Leuschner were : — 



I. The restriction of the number of the observations to three, the 

 minimum number necessary for the solution of the problem. 



II. The reduction of the fundamental data to be approximated. 

 These are (a) two observed geocentric coordinates, their velocities 

 and accelerations ; (b) the rectangular heliocentric coordinates 

 and their velocities — all for the normal date. 



III. A short method of obtaining preliminary values of these 

 quantities which will do away with the solution of simultaneous 

 equations, as in Harzer's method. 



IV. A short method of solving Harzer's equation of the seventh 

 degree for the geocentric distance. 



V. A shoit method of determining the final values of the funda- 

 mental quantities (6) from which the elements are computed. 



To ascertain how these improvements are effected reference 

 must be made to the volume itself. Suffice it to say here that the 

 solution of the equation of the seventh degree is accomplished by 

 showing that it reduces to the form of an expression on which 

 Oppolzer bases Table XIII a of his Lehrbuch zur Bahnbestimmung. 

 In the work before us Oppolzer's table has been extended to six 

 places of decimals by Prof. Leuschner. It appears there as 

 Table XVI, covering 42 pages, and by means of it the necessity 

 for solving Harzer's equation by trial is avoided, and the geocentric 

 distance can be found directly by interpolation. 



