142 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



which on reduction gives 



Thus P" n is a solution of the equation (3) ; the form (2) shows that it does not 

 become infinite when p = 1> while from (3) we see that the two functions 

 P*, cos s<j), P' sin S(f>, or what is equivalent, the two functions P^e*'"*, are spherical 

 surface-harmonics of order n. In our subsequent work the latter forms involving the 

 imaginary exponential will be more convenient than the real trigonometrical forms. 

 We shall therefore describe the functions P' (/u,)e "* as the tesseral harmonics of 

 order n and rank *. In some cases it may be convenient to apply the same nomen- 

 clature to the "associated function" P?, (p), but whenever this is done, it must be 

 understood that an exponential factor is implied, though not expressed. 



The tesseral harmonics of course include as special cases the zonal harmonics 

 obtained by putting s = 0, and the sectorial harmonics obtained by putting s = n, 

 while, in accordance with the definition (2), for values of * greater than n we may 

 suppose that P*,^) = 0. 



The principal properties of the tesseral harmonics which we shall require may be 

 derived from those of the zonal harmonics. Thus, if we differentiate s times the well- 

 known relation 



(n + 1 ) P n+I - (In + ]) ,iP, + nP B _, = (4), 



we obtain 



" d^ dp 



which, on making use of the formula 



gives 



On multiplying by the factor (1 /x 2 )= J this may be written 



(n-s+ 1)P; 1+1 - (2n + 1) M P; -|- (n + s)IV, = . . . (6). 



Again by difierentiating the equation (2) we find 



