Mr. A. Cayley on the Theory of Logarithms. 355 



i. e, we have (iioTi g'f Q'ii:<J^isq(u9J Siii abiBWi?)Jiii .^aoimoipt 



^•^u,! lu, i:jv-\^ yilOV- 0?= +,' €=±0, ' '*' ^^^ \(>k{liVj OV/ „ '•• 



And of course the other logarithms in the equation have an ana- 

 logous signification. 



Hence, attending to the equation 



- - .. ' tan'V^^ta»-^<*^tan-^^^^^4-e^'^^^^ 



where^ when l+«^ is positive, e"' is equal to zero; but when 



1 + ay8 is negative, e'" is equal to + 1 or — 1, according as /3— a 

 is positive or negative, or what is the same thing («, ^ being of 

 opposite signs when 1 + a/S is negative), 



we find 



where e, e', e'', e'" are defined by the conditions — 



0!=+, 6 = 0, _^^ , . ^ ■ _ 



,A V.' X ar ;^'' Tn| v-rrv: '- -;•'-': ■a?TO00i5 3d:? 



«'"^''' ..yl_j„_,.^^ y-y'- v ^'"^ 



X X X X X 9i 



Suppose, to fix the ideas, that ^, y are each of them positive, 

 we have 6=0; and considering the several cases,— 



Here a?a?' + 2/2/'^ +, 1+ — ^ = + ; and consequently not 



X X 



only €^=0, but also 6"=0, €'"=0, and thence E=0^,^ ^^ ^^^m^ 



Here €'=0. Moreover, xx' being positive, dex^-hyy^ and 14-^ 5^^ 

 will have the same si^o. If they are both positive, then e^'ssO, 



