264 DEVELOPMENT of a certain 



can, by means of them, exterminate all the ellipfes, except three, 

 and thus obtain a fingle equation, exprefling a relation between 

 any three ellipfes we may choofe to retain. 



Now, in confidering the feries of quantities e\ e\ e, e\e'\ &c. 

 which denote the excentricities of the ellipfes, it will readily ap- 

 pear, that in continuing the feries forward, they rapidly dimi- 



I — — c I — c* e 1 



nidi ; for fince e' rr ^-^r, =■ , +c ^ z =: , +c y, it is evident, that 



however fmall the conjugate axis of any one of the ellipfes may 

 be, the excentricity of the next ellipfe in the feries will be ex- 

 preffed by a fraction lefs than the fquare of that which expreff- 

 ed the excentricity of the former ; but if the conjugate axis be 

 nearly equal to the tranfverfe, then the excentricity of the next 

 ellipfe is only a little greater than one-fourth of the fquare of 

 the excentricity of the former ellipfe. 



Let us fuppofe, for example, e zz .5, then it follows that e' r= 

 .071797, e" — .001292, and e"' is fo fmall, as not to admit of 

 being expreffed by fewer than fix cyphers between the decimal 

 point and nrft Significant figure to the right hand ; and, confe- 

 quently, the ellipfe of which it is the excentricity, differs not 

 fenfibly from a circle. 



Hence alfo it follows, that the feries of excentricities <?, e\ i\ 

 &c. continued in the oppofite direction, mud approach rapidly 

 towards 1 , whichjs their limit ; and if we fuppofe e — .5, as be- 

 fore, we mall have £ z=. .942809, e* — .999566 ; as to e"\ it dif- 

 fers fo little from the tranfverfe axis, that the ellipfe, of which 

 it denotes the excentricity, may be confidered as a ftrait line. 

 Thus it appears, that the rectification of any ellipfe may be re- 

 duced to the rectification of two other ellipfes, either considera- 

 bly more excentric than itfelf, or considerably lefs excentric. 



12. It is now to be obferved, that the two ellipfes by which 

 we have expreifed the coefficients of the development of the 

 formula (a* -f- b"- — iab cof <p)" f have the fame relation between 



themfelves, 



