662 PHILOSOPHICAL TRANSACTIONS. [aNNO 1702. 



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





g, &c ; then the quadrature of the curve whose ordinate is - ■ or ■ 



vrr—xx vrr + xx 



will be obtained by this theorem in finite terms. 



Corol. 1. If n be expounded by any term of the following series, ] , 2, 3, 4, 5, 

 6, &c; then the curve whose ordinate is -- . — r or .- is exactly squared 



V rr — XX vrr + xx ■' ' 



by this theorem. 



Carol. 3. If ?« be expounded by any term of the following series, 0, 2, 4, 6, 



8, 10, &c ; then the quadrature of the curve whose ordinate is - ' depends 



v?T — xr 



on the quadrature of the circle; for if with the centre c, and radius ca = r 

 (fig. 4, pi. 14) the circle aeg be described : and there be taken cd = .r, erect- 

 ing the perpendicular de, and joining ce : then the sector cae divided by ^ rr, 



x" 



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



-s/rr — xx 



with the centre c and transverse semi-axis ca = r, (fig. 5,) there be described 

 the equilateral hyperbola eam ; and having drawn cf = t perpendicular to ac, 

 also FE parallel to the axis till it meet the hyperbola in e, and joined ce : then 

 the hyperbolic sector cae divided by \ rr, is equal to the area of the curve whose 



ordinate is 



'vrr-\-xx 



Carol. 4. If VI be expounded by any number different from ihe foregoing ; 



then the curve whose ordinate is ■ . or , is neither exactly squared, 



v/T— XX vn' + xx 



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



Theoeem V. If a be the area of a curve whose absciss is x, and ordinate 



- — ; and b the area of another curve havinsr the same absciss x, but its ordi- 



d—x *" 



nate : then will a = d"B — ^ — &c. But if theordinate 



o — X m in — 1 m — 2 



«ni ^m clx^^^^ d' l^'~ ~ 



be -; — , then the area will be a = 1 ^ &c. + ^"8. 



a-f-.r 711 m — 1 »j — 2 



Corol. Km be expounded by any term of the following series, O, 1, 2, 3, 4, 



5, 6, &c; then the quadrature of the curve whose ordinate is -j — -, or /--, de- 



pends on the quadrature of the hyperbola. For drawing de, ef at right angles, 

 fig. 6, pi. 14, take eg = d, and draw gh perpendicular and equal to it ; then 

 between the asymptotes de, ef describe an hyperbola passing througli a ; then 

 take GK = X, towards e in the first case, but towards f in the second ; and draw 

 the ordinate kl : then the area hgkl divided by (Id, is equal to the area of the 



x" x° 



curve whose ordinate is - — , or . - . Hence the solid generated by a portion 



