VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS, 659 



curve to be compared with the curve whose ordinate is ar-4 */ dx — xx ; then 

 because in this case n = I, therefore a = d"n X -z — — ; r-;; x"'-^7f. But 



TO = — -t, therefore 2m +1=0, and a = ■— ar™-' y^ = — „ ^,. 



Here it may be observed, that the area thus found, will be sometimes less, 

 and sometimes greater than the true area, by a given quantity. Now that such 

 defect or excess may become known, let the area found be augmented or di- 

 minished by the given quantity q ; then putting a: = 0, let the area thus in- 

 creased or diminished be put = O, which will give the correction q. Thus, in 

 the present case there will be found q = \dV d, and therefore a = \dVd — 



Corol. 2. If n be put equal to any term of the following series, 3, 4, 5, 6, 7, 

 &c ; then the quadrature of the curve whose ordinate is x-"*/dx — xx or 

 x-''Vdx -\- XX, becomes finite, and is exhibited by our series. For instance, 

 to find the area of the curve whose ordinate is x-^*/ dx — xx ; suppose it com- 

 pared with the area of the circle, which may be called a : then m = 0, ?< = 3 ; 

 therefore A =p — q — r— s. But since the quantity 2m is infinitely little, or 

 rather nothing, and is found in the denominator of the third term by which 

 d"B is multiplied, the quantity denoted by p is infinite ; and for the same reason 

 the quantity denoted by — s becomes infinite ; and therefore the quantities a 

 — a, — R, vanish : thence p = s, and the equation divided by 



2ra4-l ''m — 1 . . , 2m — 3 (Id „ , , 2m — 3 



i!!lii X :i!!i— \ it gives d^B X -, — = -^'"-3//, or t^'-B x = 



2m+i 2ra + 2' ° 2m m -^ ' 2 



drx^~^ y^ : and writing and 3 for m and n, it gives ds X — J = 



y> ■ 2y3 



-, or B = — -r^. 



Corol. 3. If 7« be put equal to any term of the following series, — 2, — i, 

 O, 1, 2, 3, 4, 3, &c. ; then the quadrature of the curve whose ordinate is 

 x"" */ dx — XX depends on the qttadrature of the circle ; but that of the curve 

 whose ordinate is x'"V'dv -j- xx on the quadrature of the hyperbola; and the 

 relation of such curve to the circle or hyperbola, is exhibited by our series 

 in finite terms. 



Coral. 4. If m be expounded by any other number different from those 

 abovementioned, then the curve whose ordinate isx"'v' dx — xx or x'"\^ dx -\- xx, 

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

 to a simpler curve by our series. 



Theorem II. Let a denote the area of a curve, whose absciss is x, and 



4 p 2 



