148 Notions respecting New Books. 



In conclusion I cannot but think that the objections raised 

 to the former experiments were groundless, and that, when it 

 is found that a stream of acidulated water flowing down between 

 the poles of a magnet polarizes platinum plates placed in it, we 

 are entitled to call it chemical decomposition, since it would 

 only be necessary to exalt the magnet's power and increase the 

 velocity of flow in order to see, under certain precautions, the 

 liberated gases streaming from the plates. 



XIX. Notices respecting New Books. 



An Elementary Treatise on Laplace's Functions, Lame's Functions, 

 and Bessel's Functions. By I. Todhunter, M.A., F.R.S., Hono- 

 rary Fellow of St. John's College, Cambridge. Crown 8vo, 

 pp. 348. London : Macinillan and Co. 1875. 



rPHIS volume is designed as a continuation of the authors 's two 

 -*- previously published volumes on the Differential and Integral 

 Calculus respectively. Nearly two thirds of it (and these are the 

 parts to which our remarks will be restricted) are devoted to an 

 investigation of the properties of Laplace's Functions. This sub- 

 ject has been hitherto treated, at all events in Euglish books, in 

 subordination to their immediate application to physical questions. 

 Thus, they have been treated by Pratt and O'Brien with a view to 

 their application to Attractions and the Figure of the Earth ; by 

 Murphy, to the Theory of Electricity; and in the treatise on 

 Natural Philosophy by Sir "William Thomson and Professor Tait 

 their properties are discussed, under the name of Spherical Har- 

 monics, in an appendix to the Kinematical Introduction to that 

 work. Mr. Todhunter has made the properties of these functions 

 the subject of a substantial treatise, and has thereby done good 

 service to the comparatively few students w 7 ho are likely to make 

 acquaintance with a calculus " the most singular in its nature and 

 the most powerful in its applications that has ever appeared"*; 

 but whose fundamental points are hard to understand, and whose 

 applications are not without obscurity. 



If the distances of two points from the origin of coordinates are 

 r and r , and if they contain an angle ft, the distance of the points 

 from each other is proportional to (1— 2a cos /3 + a 2 )?, where a 

 denotes the ratio of r to r. "We may write this in rather a 

 generalized form (1 — 2ax + a 2 )h. If now we denote the reciprocal 

 of this expression by TJ and expand it in a series of ascending 

 powers of a, the general term will be P n», where P B or, as it is 

 sometimes convenient to write it P n (#), is a rational and integral 

 function of x. To these functions Mr. Todhunter gives the name 

 of Legendre's coefficients, and he devotes 130 pages to an exposi- 



* Airy, ' Figure of the Earth,' p. 192. 



