Mr. A. Cayley on the Theory of Logarithms. 27 



VII. A = — , a/EE— , xx' — yi/'^ + 



e'=±l=y'=-^^ 



e" =0 



e"' = ; 



and therefore e + e' — e" + e'" = if - = — ^, 



X x' 



but 6 + e'_e" + e"'=+2 = -f^ + 4)if- = ^. 



\x x'/ X x' 



VIII. x^^—, a,'=— , xx' — yy' ^ — 



e=±l=y=-'l^-h+y;) 

 X \x x' / 



and therefore e + e' -e" + e'" =±2=— f- + ^). 



Va; a;'/ 



Hence writing 



log {x + yi) + log (a;' + y'i) — log (a; + yi) {x' + y'z) = 'Em, 

 we have E = 0, except in the following cases^ viz. — 



1. (See IV.) x=+,x'^-,xx'-yy' = -X=-(-+-X 



x' \x x) 



whereE=+2 = ('^ + ^,V 

 — \x x' ) 



2. (See VI.) x = -, x^ = + , xx^ -yy^~-,y^-(y + l\ 



X \x x'J 



where E=±2 = f^ 4-?^,). 

 \x x'J 



3. (See VII.) x=-, x'~-, xx<-yy<=+, |=^, 



where E= ±2 = - f?^ + ?^,V 

 \x x'/ 



4. (See VIII.) xEE~, x<~-, xx'-yy'= -, 



where E=+2=-f^H-?^V 

 ~ \x x'J 



