CHAPTER V. 



IRROTATIONAL MOTION OF A LIQUID I PROBLEMS IN 

 THREE DIMENSIONS. 



82. OF the methods available for obtaining solutions of the 

 equation 



V 2 < = 0), 



in three dimensions, the most important is that of Spherical 

 Harmonics. This is especially suitable when the boundary condi- 

 tions have relation to spherical or nearly spherical surfaces. 



For a full account of this method we must refer to the special 

 treatises*, but as the subject is very extensive, and has been 

 treated from different points of view, it may be worth while to 

 give a slight sketch, without formal proofs, or with mere indica- 

 tions of proofs, of such parts of it as are most important for our 

 present purpose. 



It is easily seen that since the operator V 2 is homogeneous 

 with respect to x, y, z, the part of c/> which is of any specified 

 algebraic degree must satisfy (1) separately. Any such homo- 

 geneous solution of (1) is called a 'solid harmonic' of the algebraic 

 degree in question. If <f> n be a solid harmonic of degree n, then 

 if we write 



$n=r n Sn (2), 



* Todhunter, Functions of Laplace, &c., Cambridge, 1875. Ferrers, Spherical 

 Harmonics, Cambridge, 1877. Heine, Handbuch der Kttgelfunctionen, 2nd ed., 

 Berlin, 1878. Thomson and Tait, Natural Philosophy, 2nd ed., Cambridge, 1879, 

 t. i., pp. 171218. 



For the history of the subject see Todhunter, History of the Theories of Attrac- 

 tion, c0c., Cambridge, 1873, t. ii. 



