132 
PROFESSOR C. NIVEN ON THE CONDUCTION 
or by 
n=a 0 P m *(Q-a 1 V m " + *(Q+ . . . + (-l)'aJV +3r (£) + • • • 
n=b 0 P„r +l -b 1 P m m+ *+ . .. 
• . (38) 
according as n—m is even or odd. 
m 
The function P„/'(£) maybe written (£ 3 —l) 2 T,//'(0 and we know that 
(n + m)! 
and 
— 2 ’- 
= 2 '' 
2 “m! 1.3.5 . . . (2n-l) 
(m + l)(m + 2) . . . (m + r) 
(2m+2r+l)(2m+2?’+3) . . . (2m + 4r—1) 
(m +1) (m + 2) . . . (m + r) 
(2m + 2r + 3)(2m + 2r + 5) . . . (2m + 4r + l) 
, n=m-\-2r 
n — m-\-2r J r 1. 
This relation enables us to find the constant factor by which f>(£) differs from n(f). 
For when £ is infinitely near to unity, 
m 
while { 1 (f) then reduces to 
ft of "(-!)" 
1.3 .. . (2??i + l) 
Vc6 0 f™( —l)'" i+1 
1.3 .. . (2m+ 3) 
, n—m even 
, n—m odd, 
. ), n—m even 
. . ), n—m odd, 
terms above f m+1 being neglected in both sets of formulae. 
Hence if 
fl(f) = K.fl(£), n—m even [ 
= K'fl(£), n—m odd, f 
(39) 
(-l)*h , a 0 K=1.3 . . . (2m-\-\) 
2(m+l) , 2 2 (m + l)(m + 2) 
a ° 2 m + 3 ° l + (2m + 5) (2m + 7) °' 2 
2 3 (m + l) (m + 2)(m + 3) 
(2m+7) (2m+ 9) (2m+11) a 3 + ' 
/ 1 w+i\ ni+Um+ 11 , K'—1 3 (Vm-\-3)[b _ 2 ( m +l) 7 , 2“(m + l)(m + 2) , 
(-1) \ c D 0 K_1.3... (2m+8) O 0 2m+5 °i + (2m + 7)(2 m + 9) 
r* (40) 
(2m+ 7) (2m+ 9) 
2 3 (m + l)(m + 2)(m+3) 
(2m+ 9) (2m+11) (2m+13) 
^ 3 + • • • 
