238 A NEW and UNIVERSAL SOLUTION 



wherefore p' — <p — $ n: I 73° 54 36", which is greater than 

 the anomaly of the eccentric, reckoning from the aphelion, but 

 very near to it : the error not exceeding three or four feconds. 

 The eccentric anomaly, reckoned from the perihelion, is there- 

 fore nearly 6 ° 5' 24*, but greater than this arch. 



16. In this method of calculating, though any degree of ac- 

 curacy may be obtained, yet when the diftance from the peri- 

 helion is very fmall, the computation may run out to confider- 

 able length. In feeking a remedy for this inconvenience, I 

 faw that much advantage might be obtained, by compa- 

 ring the motion in an eccentric ellipfe with the motion in a pa- 

 rabola, fince at the perihelion they fo nearly agree. The refult 

 of this comparifon will perhaps be thought to make an ufeful 

 addition to the general methods explained above. 



Let there be propofed this geometrical problem : An ec- 

 centric ellipfe being given, and likewife a parabola, having 

 the fame perihelion diftance with the ellipfe ; it is required to 

 draw a radius vector in the ellipfe, to cut off a fector that fhaH 

 be equal to a given fector of the parabola. 



If this problem can be refolved, the application of it to the 

 prefent refearch will be eafy. 



Let the radii vectores that cut off the equal fectors from the 

 ellipfe and the parabola, be refpectively denoted by g and r ; 

 and let v and z be the angles that o and r make with the axes 

 of the curves, reckoned from the perihelia : then, confidering 

 the two fectors as variable quantities, we ihall have 



g? j) t z~ ri z, 

 for thefe are obvioufly the doubles of the fluxions of the two 

 fectors. 



Let a be the mean diftance of the ellipfe, and s the eccentri- 

 city : then, from the known property of the curve, 



, — a \ I ~~ e ^ 



^ ~~ 1 -f s cof V ' 



Suppose: 



