136 RELATIONS BETWEEN SEVERAL [BoOK I. 



and hence, 



This equation 10 combined with 8 gives 



r -f- r (c -\- -) cos/, y/ r / 







Putting, therefore, in the same manner as in the ellipse 



/ / r 



according as cos/ is positive or negative, we have 



8(/_i( v / c 

 [12] a = 



(c )2 



c 1 



The computation of the quantity / or L is here made with the help of the auxil 

 iary angle to in the same way as in the ellipse. Finally, we have from equation 

 XI. article 22, (using the hyperbolic logarithms), 



kt . . -. 1 G c , , C 



or, C being eliminated by means of equation 8, 

 tt ( c - -)cos/.y/r/ 



-) 21o gc . 



In this equation we substitute for a its value from 12, 12* ; we then introduce 



