90 Mr. H. S. Warner on the Connexion 



CQ=*4/2, QR=\/{xPQ=\/{x(^^?M) ) 

 CR = CQ-QR=^^2-(^- 4/^TT7)\/i 



= (x + ^lP=\)\Jz •*. C R V2 = x + Vx* - 1 



= CT + TP, 

 hence we have 



2(ACP) = log(CT + TP)=log(* + y). . . (b.) 



5. Let us now observe the variations of the quantities that 

 form this equation, as the point P moves along the curve, or 

 is assumed at different points of it. 



As P approaches A, both the area A C P and C T + T P 

 diminish till at A we have 2x0 = log 1. 



Suppose P to be taken on the branch A F, then A C P 

 and T P will both be negative, and the algebraic sum of C T 

 and T P will be their arithmetical difference, which will be 

 always less than 1 ; hence the logarithm of a number less 

 than unity is negative. 



Let us now suppose that a point on the imaginary circle is 

 taken, as at P 2 . Then 2 ( A C P 2 ) = log (C T 2 + T 2 P 2 ) ; but 

 if the arc A P 2 = 0, we have 



2(ACP 2 )=0, CT 2 = cos0, T 2 P 2 = sin0; 



but C A P 2 and T 2 P 2 being both on the imaginary plane are 

 affected with the symbol V — 1, while C T 2 being on the real 

 plane is unaffected with it; hence finally we have 



0\/ — 1 = log(cos0 -f V- l.sin0)'] / x 



or e^" 1 = cos0 + */ — I. sin J " 



Similarly, if P 2 be taken on the imaginary branch below 

 the real plane so that is negative, we have 



8 -^-i = cos fl — V — l. sin 0. . . . (c'.) 

 Let be = — so that P 2 falls on H, then C T 2 or cos 



35 



vanishes, so that 



-|V-l = log*/-l or e2 v = >/-l. . . . (d.) 



When P 2 reaches B, 



TtV— l = log(-l) or e* v - l =-l (e.) 



