176 PIERCE. 



3. To Extract the Square Root of P + jU, where P and U 

 are any Real Quantities. — Factoring out the square root of P, 

 we have by (5) and (6) 



VP +jU = ^ VP {f{h) + j gih) } , if U/P is positive, 



= ± VP \f{g) - j cj{h) ] , if U/P is negative, (7) 



where 



h = U/P. 



4. To Find sinh"^ (P + jU) where P and U are Both Positive 

 and Real. — 



Let 



sinh-MP+it') = A+jB, (8) 



then 



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

 whence 



P = sinh A cos P, (9) 



U = cosh ^ sin B. (10) 



Regarding signs, in accordance with the caption, we see that, since 

 cosh A is always positive, sin B is positive and sinh A and cos B are 

 both positive or both negative. 



Taking the sum of the squares of (9) and (10), we have 



p-ij^ IP= sinh- A cos2 B + cosh2 A sin^ B 



= sinh^ A cos2 P + (1 + sinh^ A) (1 - cos^ P) 

 = 1 - cos2 B + sinh2 A, 



whence by using (9), we have 



P2 



p2-[- ir-= 1 _ cos2 P + — -r, and (11) 



cos- B 



P2+ L^2^ 1 _^ gj„h2 ^4 _ ^— -. (13) 



sinh'' B 



Letting 



to 



1 - p^- ?: 



2 

 and solving (11) and (12) as quadratics, we obtain 



r = ^ — -, (14) 



cos 



'B = V ± \/p2+ V\ 



sinh2.4 = - r ± \/p2+ 1^2, 



