138 
PROFESSOR C. RIVER OR THE CORDUCTIOR 
This formula becomes simplified for the first of the series of roots corresponding to 
a given value of m, that, namely, for which r— 0 ; in the above expression £> r = 0 , and 
the value of k is 
h — m ( m “h i)+ 
1 
2m + 3 
e — 
Jh_ 
W,c 
T) 2 
- L u ,< 
o Ji 
e s + 
Pi TiV.o PiPz 
T) 3 
^1 ,o 
T) 3 
,o 
To reduce these expressions we observe that 
J) r , r z={n— n)(n 1 ), 
D, + 1 / = 2 ( 2 n + 3), D w =4(2n+5), 
D,_ L ,= — 2(271-1), D,_ 2>( .= -4(2n-3). 
. . 2n- + 2n — 2m~ — l 
Furthermore, since —- 7777 :- —1 
’ (2u —l)(2?i + 3) 2 \ 
4m 2 -1 \ 
(2»-l)(2»+3)/* 
» 2(4m 2 — Y)(n'— n)[n +n+V) 
° r '' r= (2n- 1)(2% + 3)(2?h - 1)(2 to' + 3)’ 
4(2m + l)(2m—1) ~ _ 8(2m + l)(2m —1)(2%+5) 
(2« —l)(2w + 3)(2?i + 7)’ dr+2 ’ r ~ (2 n -1 )(2 n + 3)(2?i + 7)(2 n +11) 
4(2m 4 - l)(2m — 1) ~ _ 8(2m + l)(2m— l)(2?i — 3) 
(2n — 5) (2n — 1) (2n + 3) 5 _ (2 n- 9)(2?i-5)(2n- V)(2n + 3) 
Wherein, as before, n=?n-\-2r. 
We may also write p in factors as follows :— 
(n—m ) (n —m — 1) fit + m) (n + w — 1) 
>= (2w-3)(2w-l) 3 0 + l) 5 
(n + 2 — m) (n+l—m)( n + m + 2)(?;. + m + 1) 
(2?i+1)(2ti + 3) 2 (2% + 5) 
Putting in these we find, for the first root of the series, 
k 0 =m(m- 1 - !)• 
2(m +1) o , 4(m + l)(2m + l) 3 
2m+3 (2m + 3) 3 (2m+5) 1 (2m + 3) 5 (2m + 5)(2m + 7) 
£ s + . . • • (49) 
and, in general, 
7 / , , 2n* + 2n-2m~-l 7 . 0 , , 7 . s , 
k=n(n+l)+ (2b _ 1)(2b+3) '-kr+k* +■■■ 
(50) 
where 
