178 PIERCE. 



7. To Find cosh-i (P + j^), where P and U are Positive and 

 Real. — 



Let 



cosh-HP + JU) = A+jB, (22) 



then 



P + jJJ = cosh A cos B -\- j sinh A sin P, 



whence 



P = cosh ^ cos B, (23) 



^ = sinh /I sin P. (24) 



The sum of the squares of these two equations gives 



P2 _^ ^72= cosh2 ^ cos2 P + sinh2 A sin2 P 

 = 1 + sinh^ A — sin^ P, 



whence by substitution from (24) and by solution of the resulting 

 quadratic equations Ave obtain 



sinh2 A= - V ±VL'2+ V\ and 

 sin2p= V ^VU^+ V\ 



These give, with choices of signs to make A and P real and satisfy 



(24) 



if V >0, sinh A = ± V2r (j{h), sin P = ±V2r /(/?), h = U/V, 

 if F<0, sinh^ = ±V-2r/(/0,sinP = =^V -2V g{h), h = U/V. 



In accordance with (23) and (24), in each line the sinh A and sin P 

 have the same sign before their radicals, and the angle P must be so 

 determined that cos P is positive. Whence 



cosh-i (P + jU) = ^ {a+j{<p + 2Tr7i), (25) 



where 

 if V>0,a = smh-H-]-V2Vg{h)}, ip = sin-M+V2T7(/j)}, (26) 



if F<0, a = sinh-M+V-2r/(/0}, <^ = sin-i{+V-2F^(/0}, (27) 



with h = U/V. (28) 



Equation {25) gives the value of cosh-i (P + jU), where P and U are 

 real, positive quantities, in terms of a and cp defined by {26), {27) and {28). 

 ip is in the first quadrant. V is defined by {14)- 



