Mff«VA. Cayley on the Theory of hogaaiithti/ii, 

 e'^'imOj but if they are both negative, then // n\ ^Wm^n^-^ 9rn* 



{8ince.«r«' = 4- ) and V";=>± 1 j^ .|> r->|i,>i 11* fSmWfnMfms^J^ 

 either case we have Er=0. "''^^'^ '^ »> -'ohr.rfoo MA arrd f^cftlf 



•)i.jHiere/«!'««:+l = y, «. 6.nf(9/l.,s\^AUo a?a/ being negative, 



yy 



'<» irijl-Rifir^ 



xat/^-yj/ and 1 + — ^ will have oppositiB signs. Suppose first 



xa^+yt/ i^ positive, then €"=0. And 1+ — ^ being negative, 



wehave€'''=+l=|^-|-~^, u e. €'"=-1. Bwiii xx^ -Vyi/ 



is negative, then €''= ± 1 = o?^'— ^y ^ ~ (^, — — J (since xoi^ 



\x X y 



is negative) = — -„ i, e. €"= + 1. And 1 + ^ ^ being posir 



tive, we have €"'=0. Hence in each case Ea=0. , 



/ , , -coiiJ^n.'A .AtiiiisJbiiij 



Here 6^== ± 1 ^ j/^ «. ^. 6'= — 1. Also xx' and y^ being each 

 negative, 0:^;^+^^ wUl be negative, and therefore ^r .; i 



.,J^ 



J. e. if 2, > 2-,thene"=-l; but if ^ < Sthen6"=+1. And 



;.3/' 



? 



a? 



1 H- - ^ being positive, ^" = 0. Hence if ^ > ^, then 



X ad 



E=l-1=0; butif ^<-^, then E=14- 1=2. 



Consider {x,y) {a^, y') ,,^f^^^.-^i 

 as the rectangular coordi- 

 nates of two points A, A'. 



In the case which has been ^j^;;^;^;;^ ;^^"^' ^^. 

 considered, the point A has 



been taken in the positive o.>dtlo>^L!. .., 



quadrant; and the prece- ^^ uf *>iIm of i 3 



ding discussion shows that . .^,,i^y4 ^/^ ,j „,, 



we have always E = 0, ex- ^y^ ^^ j^.^^^^ ^ ^i ^, ^^^^ 



cept in the case where the y^{y) 



finite line AA' meets the ,# ^, ,„b .»«il9d I M'm^ii a,i 



negative portion ot the ^ < « " 



axis of X, in which case we have E= +2. The same thing is 





