68 
Proceedings of Royal Society of Edinburgh. [sess. 
in the general fundamental theorem, viz., that which includes 
(a) and (fS) above, is more strongly induced by the elegance of the 
form of the theorem than by the mode of proof. In § 1 it stands 
approximately thus — 
( 1 + 1 ) ( 1 + 1 ) ( 1 + 1 ) 
\U 0 -t 0 t 0 -U 0 / \«q - t 1 t l -uf \u n _ x - t n M t n _ x -U n _J 
l ( 1 + U ( 1 + 1 + 1 ) 
v ^0 -Pa A-V V* 1-Pi J>i~V Kx -1 -pn-1 p„-i-x n _J 
*'*o-Po Po 
and then follows the passage containing the two deductions, viz., 
“quam aequationem etiam hunc in modum repraesentare licet : 
2. 
t a 0 t. 
t a m- 
1 Po Po Pf' * • 'P 
n — 1 
A X ixPo +1 xfP 1 . . 
Pn-l 
-1 
X Pn-l 
n — 1 
+1 
designantibus a 0 , cq , etc. /? 0 , /^ , etc. numeros omnes et 
positivos et negativos a — go ad + go . E quo theoremate 
videmus, coefficientem termini 
1 
X fo+ l X 01+ 1 X Pn-l+ l 
0 1 n— 1 
in expressione 
4+ 1 . . 
.+i 
aequalem fore coefficient termini t 0 a i t ** . . . t an ~ l 
in expressione 
— Pf° Pf 1 
A 
Pi’ll 1 ■ 
The use here of j8 0 + 1 , f3 1 + 1 , . . . . rather than the change made 
in the two special cases to the less natural /3 0 , /3 1 , . . . is worth 
noting. 
The theorems of the remaining four pages of the paper have a 
less direct bearing on our subject. 
