TABLE FOR TRANSMISSION LINE PHENOMENA. 177 



Using the proper signs before the radicals to make the expressions 

 positive as demanded by their left hand equivalents, extracting the 

 square roots and employing (1) and (2), we obtain the following 

 results 



If V>0, sinh ^ = ± V2Vg{h), cos 5 = ± V2Vf{h), 



if V<0, sinh^ = ± V- 2Vf{h), cosB = ± V- 2V g(h), 



where h = P/V. 



Having regard to the rule of signs enunciated under (10), we may 

 write 



sinh-i (P + jU) = x+j{y + 2x7?) and - x + j {t - y + 2Trn), (15) 



where 



if V>0, X = sinh-M + ^WgQi)], y = cos-H+V2V f(h)} , (16) 



if V<0, X = sinh-i {+ y/-2Vf{h)], y = cos-M+V- 2F^(A)},(17) 



with h = P/V. (18) 



Equation (16) gives the value of sinr^ (P -\- jU), lohere P and U are 

 real, positive quantities, in terms of x and y defined by {16) and (17). 

 The value of V is given by (14)- The signs in (15) are so chosen that the 

 value of y in the first quadrant is to be employed. 



5. To Find sinh-^ (P — jU), where P and U are Positive, 

 Real Quantities. — This differs from the preceding problem only in 

 the fact that sin B is negative, whence 



smh.-'^(P-jU) = x-j (y+27rn) and -x+j (?/+7r+.2xw), (19) 



with y in the first quadrant and with x and y defined as in (16), (17) 

 and (18). 



6. To Find sinh-^ (-P-jU) and sinh-i (-P +jU), where P 

 and U are Positive, Real Quantities. — These results may be had 

 directly from Sections 4 and 5 by use of the facts that 



sinh-i (-P - jU) = — sinh-i (P + jU), (20) 



and 



sinh-i (- P + jU) = - sinh-i (P - jU). (21) 



