[ *75 ] 
the whole area of the curve, whofe abfcifla is x, and ordinate 
xP — 1 2n + 2r- 
i — x \ Xx 
will be 
r ' r+ x A • n be- 
' f + r.p + r+ i ./>-pH-2 '(«) X ’ 
ing any pofitive integer, and p and r any pofitive numbers, whole or 
fractional. 
. II. 
By the- preceding article, the whole area, when the ordinate is 
— i — \P—l 2 r+aa — I . 
i sc Xx is 
Z.Z+ I (r) i . 2 ( z — i) i 
p + Z . P + Z+ I (r) p . £ + I (z) X 2 ’ 
the 
2 Z — - I 
being 
whole area, when the ordinate is i — ^ 1 x x 
I . 2 (z i) I 
p.p + I (%) X 2 * 
Likewife, by the fame article, the fame whole area is rrr 
— 5 - — r rJr — rr X A. Therefore this laft expreflion is ==: 
p + r.p+r+i ./> + r + 2 (z) . r 
z . z + i (r) I • 2 (z — i) 1 t-, ... . . 
—i ] — r — 7~\ X. - — —j rrX- rrom which equation, p and r 
p + z.p + z+i (r) p • p ~{~ l (-z) 2 .. . . . > - 5 ^ 
being pofitive, as before obferved, A, the whole area of the curve, 
whofe ordinate is i — x l \^ X x 2 \ 
1.2.3 
1 ) Xp + r ,/> + r+ r (z) 1 
“ x 7 * 
is found equal to 
r 
« w. • — » 
being any number what- 
p.p +• i.p -f 2 (r-fz) x r . r H~ 1 .(z) 
'ever. 
Confequently, fuppoflng # .infinite, we find A = the ultimate . value, 
.. . c 1 . 2 .3 (z) X/> + r./» + r + 1 (z) 1 
or limit of ; ~ H X 
2 Z 
p .p-\- I ./» + 2 (z) Xr . r+ I (z) 
Having thus obtained a general expreflion for the whole area 'of any 
s\P — * 
X x 
2 r — 1 
, and that 
expreflion 
curve, whofe ordinate is expreflfed by 1 — x 
