VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 15 



of the surfaces and solidities of solids formed by the rotation of planes ; for the 

 rectification of curves ; and for the calculation of their centres of gravity. But 

 before proceeding further you must understand, that I assume what the great 

 Newton has demonstrated, in p. 251, &c. of his Principia, concerning the 

 momentary increments or decrements of quantities, which either increase or 



n 



decrease by perpetual flux, and especially that the moment of any power a™ 



n n 



is - a A"» ' Hence the fluxion - a a™ being given, on the contrary we 



n 



may find the flowing quantity a™, 1st, by taking a out of the fluxion ; 2d, by 

 increasing the index of the fluxion by unity ; 3d, by dividing the fluxion by 

 the index so increased by unity. In what follows, the absciss of the curve 

 shall be denoted by x, its fluxion by .f, the ordinate by y, and its fluxion by i/. 



This being supposed, to proceed now to quadratures, first let the value of 

 the ordinate be obtained, by means of the equation expressing the nature of 

 the curve ; 2d, multiply this value by the fluxion of the absciss ; then the 

 rectangle hence arising will be the fluxion of the area ; 3d, find the fluent of 

 this fluxion of the area, and the required area will be found. 



Let there be proposed the equation x"' = yn, expressing the nature of any 



m 



paraboloid, in which the value of the ordinate h y ■=. xn; which being multi- 

 plied by .r, the rectangle x" i will be the fluxion of the area ; therefore the area 



m TO 



required will be x" or xy, putting y for x" . 



Again, let there be proposed the curve whose equation is x* -\- a^ x"^ =z y^, 

 which is the first of Mr. Craig's examples. Then talcing xV x x-\-aa=.y, the 

 fluxion of the area will be x x '>^ x x -\- aa. Now since this is involved in a 

 radical sign, suppose '/xa; + afl=:z; hence ar or + a o = z^, and therefore 

 ar i' = z i ; hence putting z i and z for x x and V x x -\- a a, the fluxion freed 

 from surds will be z^ i ; this reduced back to its origin will be ^ z^ ; which, by 

 restoring '^ x x -\- a a for z, gives \ . x x — a a ^ x x — a a for the area re- 

 quired, f. 



And that it may appear with what ease these quadratures may be obtained, 

 take one example more, viz. let the equation of the curve be — — = y"^ ; 



therefore « = —=.=, and --^L^ is the fluxion of the area. Suppose V x + a = z ; 



Vx+a Vx+ a 



being a good treatise on infinite series^ and other miscellaneous tracts in mathematics ; 4. A Treatise 

 on Annuities on Lives, &c. Mr. Demoivre was one of the commissioners of the Royal Suciely, who 

 decided in Newton's favour in the celebrated dispute between that great man and Leibnitz, respecting 

 the discovery of the doctrine of fluxions. 



