288 DEVELOPMENT of a certain 



But by comparing together the values of P and P', alfo of 

 Qjmd Q\ it appears that P = (i + e)?', and Qj= L (i +'& ) ; 

 fo that, by fubftituting thefe values of P and Qjn the formula 

 E — * P(i — eQJ), and remembering that e % = r^iy, we alfo 



get (i + E = \ F (i + e" — ze'Q^) ; and if, by means of this 



equation, and the equation, E' = |P'(i — *'Q[_), we extermi- 

 nate Q[_, it will be found that 



2E-(lrO E =( I - ''0 I P'. 



Let us now put E" to denote the quadrant of a third ellipfe 

 of which the excentricity is tf \ then, if we alfo afTume 



and confider that the fecond and third ellipfes are related to 

 each other exactly in the fame manner as the firft and fecond, 

 we get an equation fimilar to the laft, namely, 



2E* — (i -f- O E' = (i — rff J \ P' ; 



now fince V ;/ — i e „ , we can readily, by means of thefe two 



equations, exterminate P' ; accordingly we find 



(i+^d— ^)E— {2(1— e")+{i— r'0(i+^)}E'+2(i— <»)E'=o. 



Thus we have got an equation exprefling a very remarkable 

 connection between three elliptic quadrants, the excentricities 

 being e, <?', and e". But this equation may be rendered more 

 fimple in its form ; for if we put c for the femi-conjugate axis 



of the fecond ellipfe, we have e* ~ j-J-Ji (Art. n.) and therefore, 



i-f e" = i^ '; now 1— e'*=c\ therefore ( 1— </»)( i+0=C 1 — **Yl 

 hence alfo we have 2(1 — e'*) = (i — e")c'(i-{- c') t fo that by 

 fubftituting for (1 — *' a )(i+*0 and 2(1 -—<?' 2 J their values, 



and 



