20 Mr. A. Caylcy on a Transcendent Equation. 



or if 



B=slogtan(£ir + j0), 



then 



= - log tan (\ir -f £ui). 



And substituting for </> its value, we obtain 



gd .u= — log tan {\ir + £ui), 



which is the definition of the transcendent gd . u. It is to be 

 noticed that gd.u, although exhibited in an imaginary form, is a 

 real function of u; and, moreover, that it is an odd function, 

 viz. wc have 



gd( — u) = —gd{u), 



and therefore also 



M»)=o. 



The original equation, 



u = log tan (lir + £</>), 

 written under the form 



shows that wc have 



: =*>*(y) =*>*(— «#); 



or substituting for </> its value gd . u, we have 



u = igd( — igd ,u)j 



which may also be written 



iu=gd . igd ,u. 



So t\\&t gd.u is a quasi-periodic function of the second order — 

 a property which has not, at least obviously, any analogue in the 

 general theory. We have 



cos gd u = I {e^' d • " + e~ * e d u ) 



1/ 1 -H tan jui 1 — tan|wi\ 

 2 \ i — tan £ ui 1 -+• tan | ui) 



1 + tan 2 £wi 1 



1 — tan 2 ^ ui cos wz 



