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 Ifih) -hjg(h)}, if U/P is positive, 



= ± VP {fig) - j g{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 



smh-' {P +jU) = A+jB, (8) 



then 



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

 whence 



P = sinh A cos P, (9) 



U = cosh yl 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 sura of the squares of (9) and (10), we have 



P2-I- u^= sinh^ A cos2 B + cosh^ A sin^ B 



= sinh2 A cos2 P + (1 + sinh^ A) (1 - cos- B) 

 = 1 - cos^P + sinh2.4, 



whence by using (9), we have 



Letting 



V = V^^' (14) 



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



cos-p = r ± \/p2+ V% 



sinh^.l = - !'■ ± \/P-+ r^. 



