188 THE ROYAL SOCIETY OF CANADA 



r (v + 5) J 



TT =rw [ 3ir(44) ' *'■• ~ 2!i>+5) iv+2)Kv -'' 



etc. 

 The general form for these identities is 



25-1 



—7- = r(i>) 



[»+£.»+$. • .0+ — 5- 

 5!i>+5+i) ' z ' A "'° ~ 



2 ç — 1 



z,+f.fl+f...z/+ ^— • I 



5-i!i>+5+ ^) (t+2)X - 1 ' 



z/+f.z>+£. . .H 2~-2-l 



+ ,-2!rCr + 5+3T" " (t ' + 4) * M " 



When r is made to take the special value \, we get the following 

 expansion for sin 2s in zonal harmonics of the first kind: 



sin" = 2* . si [-- „ . 1 .P - 1 j.5P, + 



LI. 3. o... 25+1 3 . 5 . 7 ... 25+ 3 



1 5(5-1) 



!»/ 



'- 1 



5.7.9. . .25+5 • 2! 

 The special ease 5 = 2, namely 



is given as an example at the end of Ferrers's Treatise on Spherical 

 Harmonics, but without any indication as to the law of the coefficients. 



