VOL. LVIII.] PHILOSOPHICAL TRANSACTIONS. 545 



pressed by the area of the curve, whose ordinate expresses the velocity of the 

 body, while the time it has been in motion is expressed by the corresponding ab- 

 scissa. Therefore the facilitating the computation of curvilineal areas will mani- 

 festly contribute to the improvement of the doctrine of motion. Which doc- 

 trine being a branch of philosophy of no small importance, such improvement 

 will not be considered as a trifling speculation, by the r. s. 



1. Geometricians have found, that if a be put to denote the whole area of 

 the curve, whose abscissa is x, and ordinate (1 — x^)!' ~ ■ X x^'' ~ ', the whole 

 area of the curve, whose abscissa is x, and ordinate (1 — x^Y ~ ' X x*" + ^-^ - ', 

 will be = '■ — - — '- — - — ^-^ — : X a; n beine any positive mteger, and 6 



and r any positive numbers, whole or fractional. 



2. By the preceding article, the whole area, when the ordinate is (1 -\- xy- * 

 Xx^' + ^^-' is= i^±l^l-.- X iililrJ) X ^; the whole area, when 



\ 2 (z l) 1 



the ordinate is (I — x'^y — ' X xi:, " ', beine = , , i X „• 



^ ' ° p . /) + 1 (z) 2 



Likewise, by the same article, the same whole area is = 



-— "^ '-^^ — ~ — —r-. X A. Therefore this last expression is = 



p + r.p + r+l.p + r + 2(z) f^ 



Z.Z+ I (r) _ 1 .2 (z— 1) 1 „ , . , ^. ^ J u • 



; — 7 -rr-T X .—rri X 77. From whicli equation, p and r bemg po- 



p + z.p + z+\{r) p.p+l{i) 2 ^ ^ .. 



sitive, as before observed, a, the whole area of the curve, whose ordinate is 



/-, 2\B_, w ir . ■ r 1 1 i. l.2.3(r + z— \)xp + r.p+r+l(z) 



(I —x^y-' X x"^-', IS found equal to ; -^ — r"^; -. \ r, rv 



^ ^ ^ p.p+l.p + 2{r + z)xr.r+l(z) 



X TJ z being any number whatever. Consequently, supposing z infinite, we 

 find A = the ultimate value, or limit of l^l^^^^^''-±^-^] J^J X k 



' p .p+ \ .p + 2(z) X r .r + l(z) '^ 2z 



Having thus obtained a general expression for the whole area of any curve, 

 whose ordinate is expressed by (1 — x'^y ~ ' X x'^' ~ ', and that expression for 

 such area consisting of an infinite number of factors multiplied together; to 

 render the same useful in practice, some theorems are requisite for ascertaining 

 the limits of such products. The theorems which Mr. L. has hitherto been 

 able to investigate suitable to that purpose, he gives in the next two articles. 



3. The limit of 1 — m* X 2" — m* X 3'" — m" (z) X -— t". is = - X sine 



of ms; N being the number whose hyp. log. is I, and s the semiperiphery of 

 the circle, whose radius is 1. Whence, by taking m equal O, we find the limit 



ofl•^2^3^(z)x— : = 2s. 



--- 



4. The limit of dz + a xdz + a+dxdz-\-a + 2dX ^7-ni'S = 2'' * . 



2 a z 



Hence, z 4- 1 . z + 2 . z -|- 3 (z) being = 1 . 3 . 5 (z) X 2*, it appears, that the 

 limit of I . 3 . 5 (z) X -z-z is = 2"^ 



VOL. XII. 4 A 



