54tJ PHILOSOPHICAL TRANSACTIONS. [aNNO I768. 



5. Writing a, b, c, v, and e, for the whole areas of the curves, whose ordi- 



» 1 1 2 



JP"^ a^ 1 X '^ J * ^ ,-1 



nates are r, -, r, — ,, and r, respective y; we 



(1 _ x')i' (1 - x*)i (1 - *')4 (1 - *')* (1 - a»)i ^ ^ 



have, by art. 2, 



A - the limit of l-2-3(z)x4. 7.10(r) 2^«-- . 



A _ the limit ot j.3.5(,)^5.ii.,7(,) X -7-' 



*u r •* /• 1 .2.3(z) X 7 . 13. I9(z) ^^ 1 

 B = the limit of ,.3.,;,; ^2. 5. 8( j X i^' 



C = the limit of ,^' ,„ ' „ , . X = the area of the semicircle, whose radius is 1 ; 



1 3 T (2 1 z 



, r ■. r t-2-3(2) X 5.U. 17 (z) .. 1 



D = the limit of 1.3.,^,)^ J. .,. y(,) X ^; 

 E = the limit of [44^'''^'i^,l?l X '— 



1 .3.5 (z) X 1 . 7.13(2) '^ z • 



Now it appears by the above equations, that -is = the limit of.-- " .3 .y jqL ( 



„„ ,.,, i.o- 6x sine 60" 3« rri, ^ • 3*8 



X 2^"; which by art. 3, is = — : — — - = — . Therefore a is = -— -. 



' •' 12 x sine 30° 2 2 



T. 1 *u 4. B X » • ^u r t r 5.7. II .13.17.19(22) ^^ 3 



It appears also, that — — is = the limit 0^ 2.4. 5. 7- » . l O^a^) ^ sI^T".' 



which by art. 3, is ^ 3*. Therefore d is = — . 



B ^ 



It likewise appears, that 



, ,. . ^ 1' . 2* . 3' (zl 3 92K— I _,, - 3c 



B X E IS = the limit of ^^ g, ^ ~ X -^ = 3 c. Therefore e = — . 



6. Writing f, g, h, i, k, for the whole areas of the curves, whose ordinates 



J t i 3 



X^ X^ 1 X ' X "^ i- 1 11 



are r, 1, n, ;, ,, respectively, we have, by 



art. 2, 



F - the limit of i-2-3(z)x9.i5.2i(z) 3^. 



F_tne limit OI 2.5.8(2) X 5.11.17(2) ^ 2z' 



^u r •* f 1 -2.3(2) X 8. 14.20(2) ^ 3« 



G = the limit of ^:jr:^^ X ^-^-; 



,, ,. ., ^ 1.2.3 (z) X 7.13. 19(2) ^, 1 



H = the limit of 2.5.8(2) XI. 3. 5( z) X 5-2 = «' 

 ^u r •. f 1^2^3'(2) ^^ 3« 



I = the limit ot ^ ^ ^ , . , '; ^, - X -r-; 



2 . 5 . 8 (z) X 1 . 4 . 7 (z) 22 



tu r .. „r 1 .2.3(2) X 5. 11 . 17(2) ^ 3-= 



K = the limit of 2.5.8(2)xi. 7.13,2) ^ Tz' 

 By which equations it appears, that - = - is = the limit of 



1.3. 3. 5. 5. 7(2z) ,, „,, w u\. ,. n ■ 4 X sine of 90° _ 2 rp.„„ 



— =- ^ — TT— z X 3"; which by art. 3, is = — : -r— 3 = - Inerc- 



5 .7.11 . 13. J7. 19(2z) ■' ' 12 X sine of 30° 3 



fore F is = — . 



It appears likewise, that 



". .. ,. .. f 1 .3.5(z) X 8. 14.20(2) ^ 3* 



- IS = the limit of ^ ^ „ ,\' — — -— — —7-, X ^ 

 B 2.5 .8 (z) X 7 . 13. 19 (z) 2« 



