VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS. 66l 



Theorem III- Let a be the area of a curve whose absciss is x, and ordinate 

 x^i^rr — xjc ; also let b be the area of a curve whose absciss is the same x> 

 and its ordinate x'"-^"y' rr — xx; and put *^»t — xx ■= y; then will it be a = 



„ TO— 1 WJ — 3 TO— 5 ., TO — 7 . 



r^" B X — :-: X X X ; &c = p 



m + 2 m m — 2 m— 4 



L ^m-1 7/3 = — a 



m + 2 -^ 



r- TO — 1 <3 r> 



— - X — -r ar"'-3 y^ = — R 

 m m + 2 -^ 



r* TO — 2 m — 3 ,. , 



■T-5 ,,3 -— s 



X — n; X a;'"-^' 7/ = 



TO— 2 TO+2 TO 



&c. 



Carol. 1. If m be expounded by any term of the following series, i, 3, 5, 7, 

 Q, &c ; then the quadrature of the curve whose ordinate is .r'"y/rr — xx or 

 x'"\/rr + ^■^ becomes finite, and is exhibited by this theorem. 



Carol. 2. If n be expounded by any term of the following series, 2, 3, 4, 5,6, 

 &c. then the curve whose ordinate is .r~^''^rr — xj: or x~^" \^ rr -\- xx, is 

 exactly squared by this theorem. 



Carol. 3. If m be expounded by any term of the following series, — 2, O, 2, 

 4, 6, 8, &c. then the quadrature of the curve whose ordinate is .r^-v^rr — xx 

 depends on the circle ; but that of the curve whose ordinate is x"'\^rr + x.v 

 on the hyperbola. 



Carol. 4. If m be expounded by any number differing from those before- 

 mentioned, then the curve whose ordinate is x"'^ rr — xx, or x"'V^ rr-\- xx, is 

 neither exactly squared, nor depends on the circle or hyperbola, but is reduced 

 to a simpler curve. 



Theorem IV. Let a be the area of a curve whose absciss is x, and ordinate 

 ■ ■ ; and let u be the area of another curve having the same absciss .t', but 



Vrr — XX a > 



X — ^" 



its ordinate —y. : then will a be equal to 



v rr — xx ' 



„ TO — 1 TO — 3 TO — 5 m — 7 o 



,.2n B X X ' X , X > &C = P 



m TO — 2 TO — 4 TO — o 



TO 



r' . "' — 1 o 



X — x'"-^ y = — 



m. ^ 



TO — 2 TO 



T'' TO— 1 TO — 3 c 



,;-"a X — X — - x"'-^ w = — s 



TO — 4 m m — 2 "^ 



"" . .. ^ ."'«-'' 



TO 



&c. 



; ^ TO— 1 TO — 3 m—b ,|j_^ 



