660 PHILOSOPHICAL TRANSACTIONS. [aNNO 1702. 



ordinate -■ And let b be the area of another curve, whose absciss 



V rf.r — XX 



is the same as the former, but its ordinate ' . ■ ■■. Put v dx — xx = v '• 



Vox — XX ^ 



then will a be =: 



, 2m-l 2»i — 3 ^ 2;« — 5 ^, 2m— 7 <, 



^"^ X -2;;r x 2^^ x .^^4 x o;;^- ^^= ^ 



x"'-';/ = — Q 



1m- I 



X -;; — a:"'--^ V = — R 



m — 1 2m 



d' 2m— 1 ^^ 2nj— 3 „, „ 



m — 2 2m 2m — 1 



ih 2m- 1 ^, 2m— 3 ^ 2w-5 ^ , 

 m-3 2wi 2wi-2 2/n-4 ' -^ 



&C. 



The observations made on the first theorem obtain here also, and likewise in 

 the following ones. 



Corol. 1. \i m be put equal to any term of the following series, \, 4, 4, -^, f. 



&c. then the quadrature of the curve whose ordinate is - or — = , 



V (/x — XX Vdx + XX 

 becomes finite, and is exhibited by this series. 



Corol. 2. If n be put equal to any term of the following series, 1, 2, 3, 4, 



5, 6, 7, &:c; then every curve whose ordinate is - /, or ■ / is squared 



' ■' Vdx — XX Viix + xx ^ 



by this series in finite terms. 



Corol. 3, If m be expounded by any term of the following series, O, 1,2, 3, 



4, 5, 6, 7, &c; then the curve whose ordinate is jr ~ depends on the qua- 



drature of the circle ; but the curve whose ordinate is 7==^ on the quadra- 



■vdx — ,t\r 



ture of the hyperbola. For if with the centre c, fig. 4, pi. 14, and diameter 

 AB = d, there be described the circle aeb ; and there be taken ad =1 x, erect- 

 ing the perpendicular de, and joining ce : then the sector aec divided by ^dd, 

 is equal to the area of the curve whose ordinate is In like manner, 



' Vdx — XX 



if with the centre c, fig. 5, and transverse axis ab = d, there be described 

 the equilateral hyperbola ae; and there be taken ad = x, erecting the per- 

 pendicular DE, and joining ce : then the sector ace divided by -^dd, is equal 

 to the area of the curve whose ordinate is -T-i^-^;^ . ' 



Vdi + XX 



Corol. 4. If m be put equal to any term not included in the foregoing limi- 



tations: then the curve whose ordinate is , , ^ :, is neither squared ex- 



Vdx ± XX ' 



actly, nor depends on the circle or hyperbola, but is reduced to a simpler curve. 



