310 FHILOSOPHICAL TRANSACTIONS. [aNNO 1717» 



in c, in a given ratio, or so that it may be bo : bc :: 1 : n. Then are to be con- 

 structed the trajectories ebp, cutting the former curve ab at right angles." 



First part of the Solution, viz. tojind the Curves abd to be cut. — 1. Drawing 

 the ordinate bh perpendicular to the axis ag, make the absciss ah = z, the 

 ordinate hb = x, and the curve ab = v : then, by the direct method of fluxions, 



BC ^ ^.r; and, v flowing uniformly, bo = — . Hence, by the conditions of 



vx 



-2) : BC (^' 



the problem, bo (-rr) : bc (-r) :: 1 : n; therefore zx •— nz.v = o. 



2. Comparing this equation with the second formula of fluxions, at the end 

 of prop. 6 of the method of increments, it gives zx-" =: vx''"\ a being a given 

 line, by the value of which the curve abd may be accommodated to any condi- 

 tion annexed to the problem. 



3. For t; writing its value v i*^ + i^ the last equation gives z = — ;==. 

 Hence will be known z from x being given, by the quadrature of the curve to 



x^ 



the absciss x and ordinate 



4. Let <r and t be integer numbers, either affirmative or negative, such as 

 that the simplest of the curves produced in this manner, may be that whose 



I— «+2!r« "^"i 



absciss isy, and ordinate y ^^ X « — y\ ; then it will be the simplest of 

 all the curves, by the quadrature of which the absciss z will be given from the 

 given ordinate x. 



5. The curve abd is always geometrical, when n is assumed equal to the 

 reciprocal of any odd number. 



6. Hitherto we have considered the curve abd as concave towards its axis ag, 

 in which case the greatest ordinate x is equal to the given right line «, which 

 may be conveniently called the parameter of the curve. And in this case the 



curve actually meets the axis. Hence the fluent of -- r^ — " , properly taken, 



viz. so as that z and x may vanish together, the curve will pass through the 

 given point a, as the problem requires. 



7. But if a curve abd be required, which may be convex towards the axis; 



then in the same manner we shall come to the equation i = — r ; which 



may also be derived from the former equation by changing the sign of n. 

 And in this case the curve abd is geometrical whenever n is assumed the reci- 

 procal of any even number. In which case the least ordinate x is equal to the 

 parameter a; and therefore the curve no where meets the axis. Consequently 

 the curve is limitted to the former case. 



