136 
PROFESSOR C. NIVEN ON THE CONDUCTION 
The effect of having only non-adjacent terms is that no <£ occurs in a higher degree 
than the first, as would otherwise he the case. 
The equation which determines k is therefore 
• • • <f>M— ^ooSi + ^cA— • • -) = °. (43) 
If we neglect e altogether, the several factors reproduce simply the values of k given 
by k=n(n-\- 1), r= 0, 1 . . . oo. 
Before proceeding to approximate more closely to these roots, I shall state two 
properties of these sums which will present themselves in the reductions, the proof of 
which is so simple that it is scarcely necessary to dwell on it— 
,-S ] =,_ 1 S r 
Pr 
</)'—1 </>' 
rS.,— , )■_1&1T--i ^3 
Yr-lVh- 
Pr-\Pr 
These properties will be of great service in proceeding to a second approximation, 
and hr calculating the coefficients. 
Let us concentrate our attention on the (r+l) tt root, for which an approximate 
value is given by <£,.= 0, viz., 
k=n(n-{- 1) 
2 n 2 + 2 n — 2m~ — 1 
(2n — l)(2?i + 3) 
(44) 
The equation may be written in full 
1 —e 
Pi _ 
0U01 
lh 
0102 
Pr 
Pllh 
PlPi 
_0O010203 00010304 
+■•..+ 
Pr 
P> 
'•+1 
0 '-- 10 ' - 
Pi 
+ • 
0<-l 0<' 0'0<-+l/ \ < />0^ > l 
Pr +1 
4 > ‘4 > ‘+1 
f • • • + 
Pr -1 
0'’—20'- 
. =0 
it being observed that no two consecutive p’s can occur in the products. 
Multiplying the equation by 0,., it becomes 
We see from this that up to e 3 the value of 0,. will be 
Wherein and 0 ',.+1 represent what 0 ) _ 1 and 0,. +1 become when we substitute for 
k its first approximate value (44). 
