174 
MR. GRAVES ON A RECTIFICATION OF THE 
consequently it is impossible to develop f _1 0 according to the ascending 
integral powers of 0. Let us then proceed to develop according to the 
ascending powers of 1 — 0 f c ; ( c being a constant, and introduced — be it 
remarked in advance — on account of the power it possesses, if properly chosen, 
of rendering the intended development of f “ 1 6 convergent.) 
To effect this purpose, let 
1 — flfc = m [14] 
Hence 
6 = (l — w)(fc) -1 ; or since, by [8], (fc) “ x = f — c ; 0 = (1 — «)f — c : 
Accordingly, after substituting in [13] (1 — co) f — c for 6, and therefore 
— co f — c d u for d 6, we find 
df -1 {(1 -»)f -c} 
d w 
V -1 (1 — w )- 1 
Hence, continuing to derive the successive differential coefficients, we obtain 
d n f _1 {(1 - »)f-c} 
n 7 
- V — 1 . 1 .2:..n — l .(l — «)- 
Hence, evidently, 
P — 1 
d f {(l-w)f-c} 
A 71 
d co 
) 
V — 1.1.2. ,.n — 1 
[ 15 ] 
(by the notation j 
d n f -1 {(l — co) f — c] 
dco n 
-d w f ^(l-aPf-c} 
. dV* 
acquires, when co — 0.) 
j being designated the value which 
Also, by [o], (V 1 {(l — w) f — c}) or f 1 f—c = 2in — c 
But, by Maclaurin’s theorem, 
^ d“f~* j(l 
f-' {(l-»)f- c] = (f -1 {(1 «* ••• 
Substituting for the successive terms of this equation their values derived 
from [16] and [15], we obtain 
{(1 — «) f — c} — 2 in — c + — 1 (co. . . + — - • • • ) 
[* 7 ] 
