304 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Introducing the Eulerian angles ©, "P, we may write * 
© © ^-<I> © + © *I> + <E> 
x = p sin sin — ^ — , y = p sin cos — , z = /ocos^sin — , s = pc os ^ cos — y— ■ 
We then find that if © = 2u and k^m we have j* 
L*' m = ( - l)fe+m ( n + &) ! ( n - m ) ! P 1 ' 1 , g+^+im$/ sin M \*-m/ cos w \2n+m-* 
,l (to + m)\{n — k\(k — m ) ! 7 v 7 
F( - to — m, k-n, k — m+X, — tan 2 to) 
= ( - 1)” ■ e + ^)!p-" . +«*+*»*/; u yn-m-^ u)m +b 
v 7 (re-£)!(m + *)! v ’ K ' 
F(m - to, k-n , & + m + 1, - cot 2 to). 
The second expression requires modification if m + & is negative. Other 
expressions may be obtained by using the identities 
F( — to - m, k - n,k - m+\, — tan 2 to) = (sec u) 2n+2m F ( — n — m,n — m + l,k - m + l,sin 2 to) 
F(m — to, & — to, k + m + 1, — cot 2 to) = (cosec TO) 2w_2m F(m — to, to + m + 1, k + m + 1, cos 2 to) 
On the other hand, if m^k we have 
. 2tt 
L w ’ w = — T_ e +tA*+m$/ s i n TO) m_ *(cos to) 2w ’ , ’* — m F (?to — to, - k - n, m-k + 1, — tan 2 to), 
(m — k)\ 
while the second expression is the same as before. It should be noticed 
that 
L-'fe ■I'isfifef c‘< o. 
This relation corresponds to the law of reciprocity discovered by Leahy and 
Schmidt. We also have the relation 
-k,— m 
( - *.+ y. -?,+*)=(- 1 )" +t f 8 1! ; I” + m . ; y, z, s), 
(to + k)\(n- m ) ! 
which becomes of interest when combined with the equation of Rodrigues j 
p;V 0 ')=(-i)* ( M|p:(co S 0). 
In the particular case considered by Schmidt the transformation of 
co-ordinates is 
or 
X = y sin cr - a COS cr, Y = /3, Z = a sin cr + y COS cr, R = r, 
sin O' cos cf> = sin cr cos 0 - sin 0 cos cr cos c fi, sin O' sin <j>' = sin 0 sin c f> 
cos O' = cos 0 cos cr + sin 0 sin cr cos c />. 
* Whittaker’s Analytical Dynamics , p. 11. 
t This formula was needed in an investigation on the Diurnal Variation of Terrestrial 
Magnetism, made during the summer of 1915 at the Department of Terrestrial Magnetism 
of the Carnegie Institution of Washington. 
f Memoire sur V attraction des spheroides , Paris (1816). 
