312 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I7. 



zx — zx X cf"' = raf^ is a fluxional equation of the first order belonging to 

 the required curve ebf. 



13. And in this equation d is the value of the ordinate bh, when the point h 

 falls in the point a. 



14. It will not be very easy, while n continues to be general, to bring this 

 equation to fluents, or to the quadrature of curves. But the points of the 

 curve EBP may be conveniently found by the description of the curve abd, 

 and of a certain geometrical curve; by geometrical, understanding a curve hav- 

 ing in its equation no fluxions, nor fluents in the indices of the powers. For 

 let the curve abd, whose parameter is a, be cut in b by the geometrical curve 

 whose equation is da^x^ — zd^xf= xd^^V d^ — x^\ then that point of intersec- 

 tion B will be in one of the trajectories sought, viz. which passes through the 

 point E ; ae being = a, and perp. to ag. 



15. Hence, if abd be a geometrical curve, then ebf will be geometrical also. 

 Scholium. — The equation kx — tJc X d"-^ •=.rod^ may be found in another 



way. For by a certain analysis, which at present I think fit to conceal, I have 



found the equation -= -: . This beine: compared with the equation — = — . 



* a zz+xx or ^ a» r 



(in art. 9) by eliminating a and a, we at last arrive at the foregoing equation 

 ix — zx X d""^ = raf. 



Example. — A very simple example may suffice to prove the truth of this 

 splution. Thus make w = 1, in which case abd is a semicircle on the diameter 

 AG, and EBP also a semicircle on the diameter ae. Now in this case 



/ = . Hence, m art. 3, it is 5:= . „ : therefore z = a ■— 



V c^ — x^, an equation to the circle on the diameter ag = a, as it ought to be. 

 Again for n writing 1, the equation zx — zx X d"'^ =: rx^ (art. 12) becomes 

 kx — zx =■ rx. Hence, exterminating r by help of the equation rr = .r.r -f kz, 

 it is *"^~'"' zzz — x\ therefore reverting to the fluents, — = — x •\- a, an 

 equation to the circle on the diameter ae = a, as it ought to be. 



Of a Roman Inscriptiorif lately dug up in the North of England. By Chr, 

 Hunter, of Durham, M. D. N° 354, p. 701. 



The inscription represented fig. 8, pi. 8, was dug up 2 years since in the 

 Roman castrum, near Lancaster; it is very legible, and gives reason to hope 

 that a search after the first fortifying this place will not be unsuccessful, espe- 

 cially, being able to fix the time of Gordian's repairing this fortress to the 

 243d year of Christ. We may reasonably ascribe its foundation to the prudent 



