576 PHILOSOPHICAL TRANSACTIONS. [aNNO 3 789. 



shall here subjoin one more to those already given in this paper. — viz. Exam. Any 

 curvilinear area abc, &c. may be supposed to consist of evanescent areas EFe/^ of 

 which the base ep is the arc of a circle, whose radius is pe = \/ (z'^ -|- y"^) and 

 sine ED = y, and altitude vf, and consequently the fluxion of the area = f/ x 



arc A of a circle whose radius is pe and sine ed=— Xi^XA= —--'^-f^~ x 



A = V5 the fluent of v contained between the 1 values of z which correspond to 

 2 values of 2/ = O, will be equal to the fluent of z/i contained between the same 

 2 values of z. 



1. From a similar method may be deduced equalities between other like 

 fluents ; for the curve may be supposed to consist of other similar curve surfaces 

 equally as circles, and the solid of similar segments of other solids equally as 

 spheres. 



3. From the same principles may innumerable serieses equal to each other be 

 deduced ; for by difl^erent converging serieses find the sum of the same quantity 

 or quantities, and there will result serieses equal to each other : for instance 

 (fig. 9), if the time of falling down the arcs ac and bc, and their interpolations 

 from the principles delivered in the Meditationes Analyticae, of which the differ- 

 ence let be D ; find the difference between the times of a body's falling through 

 BC when it began to fall from a and from b by a series proceeding according to the 

 dimensions of ab = 0' a small quantity ; and find, by a series of the same kind, 

 the time of falling through ab ; the sum of these 2 serieses will be equal to d. 

 Similar propositions may be deduced from fluxional equations. 



4. In some cases the ratios of the times of bodies falling through some parti- 

 cular distances to each other may be easily known ; for instance, let the force 

 vary as the m — 1 power of the distance ar, and a be the distance from which the 

 body began to fall ; then the velocity varies as -/(a'" — x'^)^ and the increment 



of the time as -77-—^ — ^ ; but if the parts of different curves are proportional, 



then will a, x, and x vary in the same ratio as each other, and consequently the 

 time through proportional parts of the distance will vary as a^-i"*; and if the 



bodies be resisted likewise by a force which varies as the power of the ve- 

 locities, then the times through proportional parts will vary as before, that is, as 

 a^-i*", where a denotes the proportional distances from the points where the 

 forces and resistances are equal. 



Prob. 4. — 1. Fig. 10. Given an equation expressing the relation between the 

 2 abscissae z = ap and x = p/>, and their correspondent ordinates y = pm of a 

 solid ; to find its solid contents contained between 2 values of its first abscissae 2. 

 Assume 7. as an invariable quantity, and from the equation resulting find the 



