1888.] Harmonics of the Second Kind in a Finite Form. 295 
the lowest negative power of mw that occurs in D°Q, (u) will be 
p-@*s+), Hence in (w®—1)* D§Q (uv) the lowest negative power 
of w that occurs is «~"**~!, therefore since s is never greater 
than n, (u’—1)*D°Q,(u) only involves negative powers of yp, 
and hence 
1 1 ) 
= = S$ fs 
fi’ = terms of (uw? —1)°D*P, (u) (+ 3u0t yer os hes 
involving positive powers of w and constant terms. 
This determines Rf’, and the corresponding associated function 
of the second kind is given by equation (5). 
Example.—Suppose n= 4, s =2, then 
(Ge = Dy EE, yo 8 (ee (ape = ID 
oe 15 (u* — Qu? + 1) (Tp* — 1) Ctgetsot ~) 
= 4(105u? —190u? + 81) ; 
therefore (uw? — 1)?" D? Q, (w) 
_ 1035p" — 190p° + 81 
= 15 (Tu? —1)( —1) coth*p Ie) 
3. It is however unnecessary to go through the labour of 
obtaining the value of (w*—1)*DsP,(u) by the method of the 
last paragraph. 
For it is readily proved that 
(u?—1)*D*P,(u), and (u®—1)*D°Q, (u) 
are integrals of the equation in z 
(1— #) e+ 26-1 we ——+ (n+s)(n—s+1)z=0, 
hence by solving in series or otherwise 
2 (n + s)(n+8—1) a 
s § — NTS __ pe” 8 
(w’ —1)°D Ege (a) = Al \! 2. (Qn—1) +8-2 
(n+ 8) (n+s—1) (n+s8— 2) (n+s—38) ioe { 
4 (2n — 1) (2n —3) fe ais 
+ 
where A is some constant, and from the coefficient of w”** we find 
that the proper value is 
_ 2n(2n—1).. ars eb) 
Q" 7! i 
we 
