And also 



Table of Zonal Spherical Harmonics. 515 



^ = l=.Po, 



^fpa+fp, 



^«Jp 6+ §p 3+ ?p, 



Cos 6»=P! 



Cos20=|p 2 -gP o , 



0os^=|p 4 -gp 2 -ip , 



Cos50=^P 5 -§P 3 -ip. 



There is an easy rule for expanding any function of 6 in 

 terms of P , P 1? P 2 &c. 



The following table up to P 5 was primarily calculated 

 separately by Messrs. C. E. Holland and P. E. Jones, who 

 checked each other's result by comparison. 



Mr. C. Gr. Lamb applied certain checks to the results of 

 Messrs. Holland and Jones, and then calculated P 6 and P 7 . 



It is quite easy to extend the table to P 8 , for it can be 

 proved that there is a law connecting three consecutive har- 

 monics, say the (n — 2)th, (n — l)th, and the nth. 



nY a = (2n-l)^P w _ 1 -( % -l)P w _ 2 . 



2M2 



