273, 274] Spherical Harmonics 235 



and obtain the function in the form 



I 



(1 -^'G* 1 -!)" ............ (194). 



2 



It is clear from this form that the function vanishes if ra + n > 2n, i.e. if 

 m>n. It is also clear that it is a rational integral function of sin 6 and 

 cos 0. From the form of Q n (p), which is not a rational integral function of /z,, 

 it is clear that Q?(/A) cannot be a rational integral function of sin 6 and 

 cos 6. 



Thus of the solution we have obtained for S n , only the part 

 P (/A) (G m cos mc/> + D m sin ra0) 



gives rise to rational integral harmonics. The terms P(/z)cosra0 and 

 P (//,) sin ra0 are known as tesseral harmonics. 



Clearly there are (2n + 1) tesseral harmonics of degree n, namely 

 P n (p\ cos PI 04 sine/) Pi O)... cos n0P(/A sin n0 P (,z). 



These may be regarded as the (2w + l) independent harmonics of degrees 

 of which the existence has already been proved in 239. 



Using the formula 



and substituting the value obtained in 247 for ^(/^) (cf. equation (155)), 

 we obtain P% (/JL) in the form 





_ --- 



2 n n \(n-m)\ 

 (n - w)(n - m - l)(n - m - 2)(n- m - 3) n _ m _ 4 ^ _ 



The values of the tesseral harmonics of the first four orders are given in 

 the following table. 



Order 1. cos 6, sin 6 cos <, sin 6 sin 0. 



Order 2. ^(3 cos 2 0-1), 3 sin cos cos 0, 3 sin 6 cos sin $, 



3 sin 2 cos 20, 3 sin 2 sin 20. 



Order 3. 1(5 cos 3 6 - 3 cos 0), f sin (5 cos 2 - 1 ) cos (/>, 



f sin (5 cos 2 - 1) sin 0, 15 sin 2 9 cos cos 20, 

 15 sin 2 6 cos sin 20, 15 sin 3 cos 30, 15 sin 3 6 sin 30. 

 Order 4. J (35 cos 4 (9-30 cos 2 + 3), f sin (7 cos 3 - 3 cos 0) cos 0, 

 | sin (7 cos 3 - 3 cos 0) sin 0, -^ sin 2 (7 cos 2 0-1) cos 20, 



J^ sin 2 (7 cos 2 - 1 ) sin 20, 1 05 sin 3 cos cos 30, 

 105 sin 3 cos sin 30, 105 sin 4 cos 40, 105 sin 4 sin 40. 



