198 MR. ARTHUR SCHUSTER ON THE 
R is expressible in the form 
cr=co 
R = 2 {p/ sin (cr\— a.) + q/ sin (crX + a)} Q/..., 
«r=0 
or, admitting negative values of cr, more conveniently by 
R = S PnQjn sin (crX —a). 
<T= — CO 
In the ultimate result we return to the q coefficients through the relation 
Pn~ a = ("I Y + l 
[n — cr )! 
Equating the factors of Q/ cos cr (X — a) in (14), we now find 
O-Pn = // + ?’/• 
The calculation of r/ in its present form involves the summation of the expressions 
(38), (39), and (40), the k factors being substituted out of the tables previously given. 
The somewhat troublesome labour involved in this process may almost entirely be 
avoided by a transformation of expression (38), Substituting A/ from (34) into (33), 
we find 
n.n+1. k/ = E/-B/y'-C/y, 
and by means of this equation, when n+ 1 and n— 1 are respectively substituted for 
m, we obtain 
(ra + 2)(» + o- ft) J tO A v-. = c + (B'._,-B'„ 1 )y+{C",. ! +C', H )y (41), 
Zj'l'i ~r O cLit —1 
where 
e — 
n 
n + cr +1 -rig. _ (n — cr) -rjv 
(n + 1) (2n+3) ” +1 n{2n-l) ,l ~ 1 ’ 
If the right-hand side of (41) replaces (38), and is added to (39) and (40), the whole 
expression reduces to 
K = e"+ 
1 
n .7i+l 
[To-y { kP 1 + (n - cr) (n + cr + 1) k h +1 ] - o*y'it/]. 
We have, therefore, tlie following very convenient expression which allows us to 
calculate the coefficients from the previously established values of k :— 
Pn = - (ep+f?) + - l— [ly (n- cr) (n + ar+1) k/ +1 } -cryV] (42). 
cr n.n+1 . 
The first term on the right-hand side is zero, except for the cases where the type cr 
