150 Notices respecting New Boohs. 



and particularly to what may be regarded as the main difficulty, 

 viz. the theorem that " any function of \i and can be expressed 

 in a series of Laplace's functions" — a theorem of the utmost im- 

 portance in mathematical physics, and about which much has been 

 written. Mr. Todhunter gives four, and refers to a fifth method 

 of proving it. But we infer from his statements about some, and 

 from the way in which the other demonstrations are given, that he 

 is not satisfied with any of them, and that he regards a proof clear 

 of all objection as yet to seek, — an inference strengthened by a 

 casual remark on p. 259, viz. " Admitting that the possibility of 

 expansion in a series of Laplace's functions has been established, 

 we may &c." At all events, this is what he does : he gives a me- 

 thod which " is in substance frequently repeated in the writings 

 of Poisson," and remarks that though instructive it cannot be 

 considered perfectly conclusive ; he notices M. Bonnet's proof, 

 and remarks that it seems unsound ; he then gives two others, ob- 

 serving of the former that it is doubtful whether it ought to be 

 accepted as satisfactory, and leaving the latter without remark. 

 He then devotes two chapters to Dirichlet's proof, the former con- 

 taining introductory matter, the latter the proof itself; but he 

 produces this proof not, apparently, because he is convinced of its 

 cogency, but because he is " swayed by the judgment " of certain 

 eminent mathematicians. The difficulty is an old one, and seems 

 to have been first insisted on by Ivory ; and this, it must be owned, 

 is not a very satisfactory way of leaving it ; but if there is no one 

 decisive proof of the point at issue, no better course lies open either 

 to student or author than the one actually adopted. 



We have not space to notice the contents of the latter part of 

 the volume ; but we may just mention that it is devoted to two 

 subjects, Lame's functions and Bessel's functions. The former are 

 a sort of generalization of Laplace's functions ; instead of the 

 variables r, 6, <f>, they involve X, fx, i>, which are the elliptic coordi- 

 nates of a point, i. e. of a point determined by the intersection of 

 three surfaces of the second order into whose equations X, /*, v 

 respectively enter as parameters. The latter belong to an entirely 

 independent subject. If we consider the equation 



cl 2 u 1 d u ,fi n *\ _q 

 dx 2 x dx \ x 2 ) ' 



and obtain u in a series of ascending powers of x, the solution is 



Cx n 1 1 — + — -&c. I . 



1 2(2n + 2) 2.4(2?i + 2)(2n + 4) J 



Now for C use the reciprocal of 2 tt JT(n+l), and this expression 

 becomes what is called a Bessel's function, n being any real quan- 

 tity algebraically greater than — 1, and x any real quantity. 



Though the work is mainly taken up with processes and with 

 the properties of the functions under review, yet Mr. Todhunter 

 has indicated here and there some of the applications which may 

 be made of the f ormulse. Most students will find this very useful : 



