Table of Zonal Spherical Harmonics. 513 



are referred to Mathematical treatises. Many readers will 

 be satisfied with the treatment of the subject in Mr. Ferrer's 

 excellent treatise, which is, however, written only for beginners. 

 In problems on Heat Conduction (V being temperature), 

 on Hydrodynamics of incompressible fluids (V being velocity- 

 potential*), in Electrostatics (Y being electric potential), in 

 Magnetism (V being magnetic potential), and in many other 

 applications of Physics, we require to find Y a function of 

 x, y, z which shall satisfy the equation 



j?v d?v d?y 



dx 2 dy 2 dz 



,s+^ + -^=°. (!) 



or, as it is usually written, V 2 Y = . . . (1), and which shall 

 also satisfy certain other conditions. Now there are many 

 kinds of function which satisfy equation (1). The definition 

 of a Spherical Harmonic is " a homogeneous function of 

 x, y, and z, which satisfies equation (1)." 



If such a function can be found, say of the ith. degree, and 

 if we divide it by r 2i+1 where r 2 = x 2 + y 2 + z 2 , it can be proved 

 that the resulting expression will also satisfy (1), where i may 

 be a positive or negative integer or fraction. 



Now if a- Spherical Harmonic of degree i (generally called a 

 Solid Spherical Harmonic) be divided by r l , we get what is 

 called a Surface Spherical Harmonic of degree i. 



It was shown by Green that if there is a function Y 

 which satisfies equation (1) at every point of any given 

 surface, then it is the only function which satisfies (1) 

 throughout space ; and there is always a function Y obtainable 

 which satisfies (1). It is the characteristic property of a 

 surface spherical harmonic distribution of density of attracting 

 matter on a spherical surface, that it produces a similar and 

 similarly placed spherical harmonic distribution of potential 

 over any concentric spherical surface throughout space, ex- 

 ternal and internal. 



Instead of using x, y, and z coordinates we may of course 

 use **, 0, and $ coordinates. 



* When there exists a velocity-potential V hi a portion of fluid, we 

 mean that the velocity of the fluid at any place resolved in the direction 

 s is 



_dV 

 ds' 



When the motion is " rotational," as in the wheel of a centrifugal pump 

 or turbine, a velocity-potential does not exist. In any portion of a 

 frictionless fluid, if there is irrotationality, that is, if there is a velocity- 

 potential, the property cannot be destroyed. 



Phil. Mag. S. 5. Yol. 32. Nq. 199. Dec. 1891. 2 M 





