114 REPORT— 1898. 



and, when »*,=??, we have 



From the above formula; we see that, on a sphere of radius unity, 



x: = {n - m) H;r ^-m;,{i -f^)-'- H;r = (1 -f.J "f ^ 



= mfx{l -^•^)-' H™-(« + m)H;;'-', 

 also Y;;' = m{l -fx"-)-' H™ and Z:' = (». + !) H™. 



Hence /iY,T-X™ = {n + m) H"'"', 



and /iY;r + X™ = {n - m)H;r ' \ 



also (1-A'')'^Y;;' = toH™. 



From these formulae we find 



^' '■' -' ,/c^H 



and also 



f (Y:)«.+j_/x:)y,=| (._,=)('-by.,+j__^'^,(Hr).</„ 



ilso 



These definite integrals reduce to 



n{n + l)^[{H:fdfx. 

 Hence since Z^=(»i + 1)H™, we have 



\[{x:)%^ + j[(Y':yd^i + j' {Z':;fd^={n + 1) (2«+ i)j' {K':fd,x 



[1.3.5...(2h-1)]^^" + ^^- 

 Putting Jij for n in the above equations we get 



MY™-X™ = (u,+m)H;i'-\ 



A.Y™+X™ = (w,-m)H-;', 

 and (1 -/)'Y;r. = «iH;r^. 



Combining these formulae we get 



i(^Y;r-x™)(/iY--x-) + i(^Y,7+x;r)(/xY;»+X;;'j + (l-/.^)Y;:%- 



YmVm _L ViV"' 



hence j'_ X»X™c;;u+ j'_ Y-Y»c?y. = 0, 



