[ 62 S ] 
Let A E, A F, and A G , denote any three values 
of the quantity x, having indefinitely fmall equi - 
differences E F, F G ; and let EL, F M, and G N, 
(perpendicular to AG) be the refpedtive values o fy, 
correfponding thereto ; and, fuppofing EF(=FG—x) 
to be denoted by e, let c M and dN (the fuccefiive 
values of y) be reprefented by u and w. Moreover, 
fuppofing P'p and P'p 1 to be ordinates at the middle 
points P 1 P", between E, F and F, G, let the former 
(P'p‘) be denoted a, and the latter (P'p) by ( 2 ; put- 
ting AP'=a and AP'—b. Then, if a and a (the 
mean values of at and y, between the ordinates E L 
and FM) be fuppofed to be fubftituted for x and y, 
in the given quantity £>jq + Rr + S s -{-Ft, &c. and 
if, inftead of x and y , their equals e and u be alfo 
fubftituted, and the faid (given) quantity, after fuch 
fubftitution, be denoted by R'r'+ S's'-P Tt\ 
&c. it is then evident, that this quantity %)q' + Rr 
-j- &c. will exprefs fo much of the whole 
required fluent, as is comprehended between the or- 
dinates E L and FM, or as anfwers to an increafe of 
£ F in the value of at. And thus, if b and (2 be 
conceived to be wrote for x and y, e for x, and w for 
and the quantity refulting be denoted by C? + 
Rr '- f 5V+ T"t", &c, this quantity will, in like 
manner, exprefs the part of the required fluent corre- 
fponding to the interval FG. Whence that part an- 
fwering to the interval EG will confequently be 
equal to Off}' R r & C '~P ~P R r ^ * s 
Vol. 50. 4 L manifeft, 
