546 



NA TURE 



[August 9, 1906 



however, that some matters of elementary mathematics 

 could not have been omitted without detriment. Thus 

 the discussion, at the beginning of the second part, of 

 the equations of a conic, based on the definition of a 

 conic as the plane section of a right circular cone, 

 must be superfluous for a reader who is capable of 

 following the whole of the first part intelligently. But 

 the fault is doubtless on the right side. 



The whole work consists of seven parts. The first 

 deals with those portions of general astronomy which 

 are relevant to the main purpose. The chapters on 

 time and on precession and nutation seem particularly 

 clear and good. That on aberration follows the tra- 

 ditional lines of Gauss and Bessel, and criticism would 

 be out of place here. Yet the exposition of Gauss, 

 which seems to assume the apparent composition of the 

 velocities of light and of the earth as a matter of 

 course, appears to be imperiect in view of the difficul- 

 ties in the physical theory. Is it not more logical to 

 consider the apparent composition as an inductive re- 

 sult instead of the explanation of the astronomical 

 phenomena ? 



The second part contains a discussion of undis- 

 turbed heliocentric motion. Dr. Bauschinger asserts 

 (p. 170) that Lambert's equation is of little use in the 

 case of ordinary elliptic orbits. This opinion mav be 

 disputed. It is true that the development in series is 

 of little assistance owing to slow convergence, but in 

 its original form the equation can be easily solved in 

 all ordinary cases. The natural expression of the for- 

 mulae for motion in a hyperbola involves hyperbolic 

 functions. The use of these is entirely avoided, pre- 

 sumably because tables of hyperbolic functions are not 

 as a rule accessible to the computer. 



The properties of the apparent or geocentric inotion 

 are discussed in the third part. Here will be found 

 Bruns' elegant proof of the theorem of Lambert on the 

 curvature of the apparent orbit. Incidentally it may 

 be remarked that Lambert seems to have missed that 

 measure of fame to which his unquestionable eminence 

 as a mathematician entitles him. 



The longest part is the fourth, in which the various 

 methods of determining a preliminary orbit are de- 

 scribed. An excellent feature is the compendious 

 arrangement of the working formulae. This part is 

 followed by that on the adjustment of an orbit by the 

 method of least squares. In both sections numerical 

 examples are fully and clearly worked out. 



The sixth part contains the theory of special per- 

 turbations. Three methods are given, according to 

 which the perturbations can be calculated in the ele- 

 ments, or in polar or in rectangular coordinates. 

 In the preliminary chapter, on mechanical integration, 

 the usual German notation for interpolation formute 

 is employed. It is difficult to see the advantage of 

 this over the ordinary notation of finite differences. 

 The last chapter of this section brings the reader to the 

 determination of the definitive orbit. 



Here the work might have ended, but Dr. Bausch- 

 inger has added a final part, in which he investigates 

 the determination of the orbits of meteors, satellites, 

 and double star*. These last chapters are necessarily 

 brief, and it may be doubted whether, as regards unitv 

 NO 191 9, VOL. 74] 



of subject, their inclusion is justified. But that on 

 satellites is certainly valuable, especially in view of 

 recent discoveries. 



The source of the numerous theorems which are 

 met with in the work has generally been indicated, 

 but this is not always the case. Thus the theorems 

 on p. 1S4 are due to M. Radau (Bull. Astr., x. p. ii) 

 and to Mr. Shin Hirayama (Monthly Notices, R.A.S., 

 Ixii., p. 620). Such references add greatly to the in- 

 terest, but of course it is always difficult to be sure 

 that the sources are strictly original. For instance, 

 the proposition attributed (p. 131) to van der Kolk 

 was, as has been recently pointed out, previously given 

 by Whewell. There is an index at the end of the 

 volume, but it is not so complete as it should have been. 

 .^ full index of names is needed. 



\n outline of the method of Gibbs will be found in 

 Dr. Bauschinger's work, but for fuller details thr 

 pamphlet of Dr. Frischauf may be consulted with 

 advantage. The method is based on the use of a par- 

 ticular expression for the ratio of a triangle to the cor- 

 responding sector of an ellipse. The form is mathe- 

 matically elegant and the degree of approximation is 

 high, but it was thought to entail greater complexity 

 in the computations, while, on the other hand, the 

 method by itself gave little assistance when a still closer 

 approximation proved necessary. This defect was 

 remedied by Prof. Harzer. The modified method is 

 described by Dr. Frischauf in a clear and interesting 

 manner ; the practical value of his account would have 

 been enhanced by the addition of a fully worked 

 example. The pamphlet also contains a number of 

 supplementary notes to the author's " Grundriss der 

 theoretischen .\stronomie," a work of which a second 

 edition appeared in 1903 after an interval of thirty- 

 two years from its first publication. H. C. P. 



INDVCTION AND COSDUCTION MOTOKS. 

 Moteurs a CoUecteur a Courants alternatifs. By 

 Dr. F. Niethammer. Pp. 131. (Paris: L'Eclairage 

 Electrique, 1906.) 



THE title leads one to believe that the author is 

 going to deal with at least all the principal types 

 of modern alternate-current commutator motors, 

 whereas the book is practically restricted to a con- 

 sideration of the series induction and conduction 

 motors. Shunt induction motors of the commutator 

 type are occasionally touched upon, but all remarks 

 concerning these must be considered as quite 

 erroneous. Generally speaking, the number of mis- 

 takes is too great. 



In chapter i. the historic part does not deal with 

 the machines out of which those modern single-phase 

 commutator motors have been directly evolved, which 

 are afterwards considered more closely. The pre- 

 liminary consideration of some of the types now in 

 use is full of errors, and much prominence is given 

 to the least important of these types. The indis- 

 criminate use of the expression " repulsion " motor 

 leads to the usual confusion. 



In the second chapter, which is the most important 

 in the whole book, we find the author trying to 



