Mr. A. Cayley on the Theory of Logarithms. 277 



where the upper or under sign is to be employed according as « 

 and /3 are positive or negative ; or what is the same thing, 



tan~ '« + tan " '/3= tan^ ' 



l-«y3 ' ' 

 where 



l-«/3=+, 6=0, 



l-«/8=-, e=±l=« + /3 = a = /S. 

 This being premised, then writing 



log {x + yi) = log \^x^ + y^+ ( tan"' — + ctt ji 



log {x' + y'i) = log \/a;'2 + y''^ + ( tan" ' ^ + c'tt h' 

 log (a; + yi) {aJ + y'i) = log [ {xa^ —yy') + i^y' + y^') *] 



= log v/^qrp \^x'^ + y'^+ (tan-' ^^Ltf^^ + ^''tt) i 



tan'-^ +tan-'?^' = tan-'^4^ +e'V 

 ^ a;' a'a;' — yy 



we find 



log(a? + yi) + log(a:^ + ^1) — log(^ + yi) (a?' + y'i)={e + e'— e"4- e"')???. 

 Hence, considering the different cases, — 



I. ^=+, «'=+, xx' — yy'= + 



€ =0 



e* =0 

 e"=0 

 e"' = 0, 

 and therefore e + e' — e" + e"' = 0. 



II. a^= +, «'=+, xx' — yy'^^ 



6 =0 

 6* =0 



-(I 4)' 

 and therefore e + e* — e" + e'" = 0. 



