526 Sir G. Greenhill on the 



In the passage in Lord Rayleigh's manner of the tessera! 

 harmonic PJ (yit = cos#) into the Clifford function of order p,. 

 as the order n is indefinitely increased, in its representation 

 near the pole $ — on the surface of a sphere, when the- 

 radius a is enlarged to infinity, the change is made through 



4(1 -M) = hav = sin*.** = j'^ = , =4- = 4 



1 r 2 _ 1 m 2 r 



4a 2 ~4 ?* 2 ?i 5 



agreeing with the definitions of § 1, in the limit n — >oo ; 

 and the Clifford function of order p is derived from the zero 

 order by p differentiations with respect to x, as the tesseral 

 is derived from the zonal harmonic by p differentiations 

 with respect to fi. 



Take the various expressions in the recent Phil. Mag. 

 July 1919, given by Dr. Bromwhh on " Electromagnetic 

 Waves," and apply this our principle. 



Then his equation for F = F M , 



d 2 ¥ r 2 d 2 F 



^-„ (n + l)F=^=- m VF, 



d 2 F 

 in a vibration with ^- 2 = — m 2 c 2 F, . . (1) 



becomes, with m 2 r 2 = ±x, and F n = ^" +1) C, 

 our equation for 



c = tWiOO = (-£)"<¥*), . . . (3> 



and 



F = ^*C|(a?) = sin {2y/x + e) ; 



changing for divergent waves to the hyperbolic or exponential 

 form, with m 2 replaced by — m 2 . 



Thence the expression of the other functions with x— \m 2 r z 

 as variable, mr = ~ = 2 v /«,r. 



In Lamb's k Hydrodynamics/ ty n and "^ n are both included 

 in C„+|(Jm 2 r 2 ), when the phase angle e is introduced into Ci r 

 as in F ; and then his f n (z) is C„+|( — \m 2 r 2 ), for the 

 divergent wave. 



