66 Not ices respecting New Boohs. 



and since 



2R=-95in., 



« = -501v nearly. 



This is closer than could be expected considering the ex- 

 tremely rough measurement of the two commas. It will be 

 remembered that the value of a is known to be generally about 

 •55Rto-6K. 



IX. Notices respecting New Books. 



An Elementary Treatise on Spherical Harmonics and Subjects con- 

 nected with them. By the ftev. N. M. Ferrers, M.A., F.R.S., 

 Fellow and Tutor of Oonville and Cains College, Cambridge. 

 Loudon : Macinillan and Co. 1877. Crown 8vo, pp. 160. 

 npHE authors object in this treatise is "to exhibit, in a concise 

 ■*■ form, the elementary properties of the expressions known by the 

 name of Laplace's functions, or Spherical Harmonics." More than 

 two fifths of it, comprised in chapters ii. and hi., are devoted to 

 the discussion of the particular case in which the spherical Surface 

 Harmonic (P t .) is a function of p only. This function Mr. Ferrers 

 calls a Zonal Surface Harmonic ; it is the same function as that 

 which Mr. Todhunter calls a " Legeudre's Coefficient." The author 

 investigates briefly and elegantly the chief properties of P t -, and then 

 applies them to determine the potential of various forms of attract- 

 ing matter. Of these the last which he considers is the following 

 comprehensive case : — to find " the potential of a spherical shell of 

 finite thickness whose density is any solid zonal harmonic." These 

 investigations serve as a foundation for those contained in the 

 following chapters. Thus, in the fourth chapter the subject of 

 General, Tesseral, and Sectorial Spherical Harmonics is somewhat 

 briefly treated. It is well known that the general Surface Har- 

 monic of the degree i consists of 2*4-1 terms of the form 



C cos crcb sin" F?/*\ S sin c6 sin" 6> £*W . 

 r dp' r dp' 



to these terms individually Mr. Ferrers gives the name of Tesseral 

 Surface Harmonics of the degree i and order a ; and the last of these 

 terms, viz. those for which a=i, he calls Sectorial Surface Harmonics 

 of the degree i. In the fifth chapter he notices very briefly the 

 Spherical Harmonics " of the second kind ;" and in the sixth chapter 

 he treats of Ellipsoidal Harmonics, a name which he proposes to 

 give to the functions called by Mr. Todhunter " Lame's functions." 

 It is well known that one of the standing difficulties of this sub- 

 ject resides in the proof of the theorem that " any function which 

 does not become infinite between the limits of integration can be 

 expanded in a series of Spherical Harmonics." Thus, Mr. Todhunter 

 notices four or five proofs, and is not, to all appearance, completely 



