414 



Transactions. — Chemistry and Physics. 



Art. LII. — Two Spherical Harmonic Relations. 

 By C. Coleridge Farr, D.Sc. 



[Read before the Philosophical Institute of Canterbury, 3rd September, 



1902.] 



The relations proved in this paper were given without distinct 



proof as following simply from the generalised form of four 



others which I discovered in working out the expressions* 



for the intensity of the magnetic force in the interior of coils 



of various lengths. They were, however, cut out by Professor 



Lamb, who very kindly communicated my paper to the Koyal 



Society, as he did not see how they were obtained. This of 



itself causes me to think they may be new, and, as the 



original four are, I believe, new ones, hope in this direction is 



strengthened. I have known them for some years now, but 



as they follow simply from the four already published I had 



not up till now thought them worth publication. I have, 



however, been recently rather strongly advised to print them. 



Using the notation of my previous paper, the original four 



are these : — 



d Po- Id 



Pcr+1 



(1) 

 (2) 

 (3) 

 (4) 



dx r 

 _d_ 



dx 



di 



<r + l ' 



'Per 

 Po- 



r^+ 2 dO 

 d 



r o—l Per _ I 



(o- + i) ! Po- + i 



dz l r a ' +1 



• <r+i + l 



r°-P<r=(- 1) 



Po-- i 



dzi ' -- " — v -v (o--l)\ 

 Now, since the order of differentiation is indifferent, we have 

 d* d Po- d di Po- 

 dz l dx r°" +1 ~ - dx dz l r ^ 1 ' ' 



Taking first the left-hand side of the equation and making use 

 of (1) we have 



d l d Po- d i 1 d 



dz T dxr°+ l = dz 1 7^+* ~dd Fa + 1 ^ 



Now, taking the right-hand side of (5) and using (3), it follows 



d di Po- d (o- 4- i) ! Po- + i 



dx dz 1 



• 0- + 1 



dx o- ! 

 (o- + i) ! 



r 

 1 



<r + i + l 



d 



<r 



r <r + i + 2 c lQ 



— Po- + i + 1 



Roy. Soc. Proc, vol. lxiv., p. 193. 



