314 Mr . Horner's new method of solving numerical 
of the figurate coefficients in that formula of the theorem, of 
which such letter is the first term. 
14. Although this statement appears only to have returned 
to us the conditions of Art. 6 , with all their disadvantages, 
and to have merely substituted 
Aj for D<pR 7 or a! 
B 2 for D 2 <pR 7 or U 
C for D 3 <pR 7 or c' 
3 
&c. yet, by means of the property just alluded to, the essential 
data A, B, C, &c. which have disappeared, will again be ex- 
tricated. For the developement of D^pR 7 , found in Art. 8, 
undergoes thereby the following analysis : 
+ m Nr + — r 7 P/ 2 -f -r~T — — 7- Q r + 
= M 4 Nr 4 Pr 9 + Or 3 -f 
+ Nr + 2 Pr* + 3 Qr 3 4 
+ Nr + 3 Pr* 4 6 Qr 3 4 
4 Nr 4 m Pr 3 4 ’V ' ~V Q 7 ’ 3 H~ 
which equivalence will be thus expressed: 
M ffl = M4Nr4N/4Nr4 + N m r 
Returning therefore once more to our theorem, we now have 
<pR" = cpR' 4 A? 
A — (A 4 B^) 4 B 7 r' 
B 7 = (B 4 Q i r 4 -C / ) 4 C'r 7 
C = (C 4 D t r + D/ 4 Dr) 4 D 7 r 7 
. • • • • • 
V' = (V+U / +U / +U 3 r+ 
U' = (U + n-l . r ) + f 
U K _,r) + UV 
