t 43 2 ] 
XXV. A new Method of finding Fluents by Continuation. By 
the Rev. Samuel Vince, A. M. F. R. S. 
Read July 6, 1786; 
ART. I. Put F = 
M . 
X X 
n . n 
a + # 
= x m ~ n x - a n x m - ttt x + a tn x"*-s”x 
vti tn — vn . m — n m — 2 »TI m — 3 « 4 -l 
+ then F= + - * - &c. ' 
a n + x n m — nfir I w — 2 «+i m — i 
=izW, where W reprefents the fluent of the laft term. Now 
X X X xX.a -\-x 
n , n I 
l -j-* 1 
fi B n 
a + * 
; hence C. z~~= C 
n , wj c/ » , n 
a +x 
a -f* 
xT+T]' J' F x. - r - = (by fubftituting for F 
7 + 7 ) 
its 
value in the latter quantity) F x a n + a?"] 1 r x 
* y y « 4 - 1 
r *"* ^ I - r . na y p x m n x i — r . na n v 
J T7~f J m _.? >nJcl J —~y 
a +x » -p* a -\~ x * 
+ &c. 
/w«I 
— r.vx n I x . 4. 1— r.« 
tranfpofe 
n 1 j 
-pA’ 
/» — « + I 
« . 
ttt 
and divide 
a +# 
both fades of the equation by - — and we have 
4 J m—n + 1 
/ 
# x m — n + 1 
xFx 7 T 7 i t- ’ + - m ~ n+ 1 
v 1 -r .na n 
7 \ r 
rti — rn -\- 1 
m — rn -p I 
m— 2 n+i 
a -p x l 
, m - n . 
'x X 
m— n 4- 1 
zn ^ .jn— zn . 
v l-r--r-.no v rx x & 
, m—n+ 1 
n 
a -P* » 
m — rn + i 
ai - 3 " +i ^ 7^' 
w — r» 4 * 1 
/ 
