101] 
1 
155, 156, 158 
101. 
NOTES ON LAGRANGE’S THEOREM. 
[From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 37—45.] 
I. 
If in the ordinary form of Lagrange’s theorem we write (x + a) for x, it becomes 
x = hf {a + x), 
F (a + x) — Fa + ^ F'afa + &c (1) 
It follows that the equation 
F (a + x) = Fa + j ( Fa f a ) + > ( 2 ) 
must reduce itself to an identity when the two sides are expanded in powers of x; 
or writing for shortness F, f instead of Fa, fa, and 8 for ^, we must have 
1 
M 
r 8 r F=S 
$p-i fF. fp) 
L r-pf- 
—p f-p\ 
8 r ~pf 
(where p extends from 0 to r). Or what comes to the same, 
(3) 
,8 r F=S 
p[p — «R -8 [r — p] r ~P [s — l] s 
$P-sfp _ $r-pj--p _ gsp 
.(4) 
where s extends from 0 to (r — p). The terms on the two sides which involve 8 r F 
are immediately seen to be equal ; the coefficients of the remaining terms 8 S F on the 
second side must vanish, or we must have 
S 
(5)