Method of correfpondent Values , &c. 
x ?H 
sc — p . x — r . x — s . & 
— x 4- 
X- 
■p . at — q . — j . &c. 
x Sr 4- 
q — p , q — r . q — s . &c. x ’ r~p . r — q . r *-s , &c. 
&c. as in Ex. i. = Sp + (x—f) X + - 2 _ x %) + (* ~p) 
1 x — i_ x Sr) + 
r \ f 1 1 
(*“?) ( ■ x x 
P—1 t~ r 
Sfr + O— x - 
J q-p H- 
(*-P) (*“9')(*- r ) (jz 
x Sj'Hh 
I 
p-q p 
p — s 
r — p 
x Sj t> + 
r — q 
1 
q P q~ r q—s 
x Sq 4- — . x Sr + -i— . — . — x Si) - &c. 
1 1 r—p r — q r — s $ ~p s — q s — r ' 
The equality of thefe two different quantities will eafily 
appear by finding the co-efficients of both, which are multi- 
plied into the fame given value of y as S^, S^, Sr, &c. and the 
fame power of x ; for with very little difficulty they will in 
general be found equal. 
It is evident from this refolution that, giving the ordinates 
and their relpedlive diffances from each other, the value of 
any other ordinate at a given diffance from the preceding, 
found by this method, will refult the fame, whatever may be 
the point affumed from which the abfeifs is made to begin. 
3* 
i. Let a feries be A# + Bx z 4- (V + Dv* + &c. of fuch a for- 
mula that if in it for x be fubflituted a + b, there refults a feries 
A x<;/4"^d“-Bx^? 4“ b 4“ L x a 4~ b 4- D x a 4~ b 4~ &c. — (^Aa 4~ 
Ba z -}-Ca 3 4- Da 4 4 s * &c.) x (i -j-qb + rb z -\-sb 3 + tb 4 4- &c.) + 
(14-^4- ra + sa 3 + ia 4 4- &c.) x ( Ab 4- Bb z 4- Cb 3 4- D b 4 4- &c.) 
then will the feries Ax 4- Bx z 4- Cx 3 4- Da; 4 4- &c. = Ax + 
2B 2 2 . qC 3 , 24ABC — 8K 3 4 , a6C 2 A 2 + 24ACB 2 ~i6B 4 5 , 
% 4- - - ' 4~ ~ — 4" ~ x 4~ 
1.2 I.2.3 I.2.3. 4 A I .2.3.4. 5A 3 
3. 24A 2 BC 2 ~ 4 X24AB^C e 216C 3 A 3 + 4 3 2A 2 B 2 C 2 - 3 84ACB 4 4 64B 6 y 
1 . 2 . 3 . 4 • 5 . 6A 4 x % ~ 2 . 3 . 4 • 5 • 6 . yA s ^ 
4- &c. ; 
