72 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and freeing the line at A, as at A 4 , we have 



, . , cosh 8^ cosh8 B G /B G fF , , 



g =y sinhS^ • , • — r-r- • -f- • -f mhos. (137) 

 * J0 cosho £ cosh8 c Gfo G fQ 



Equivalent T. Second Method. 



Freeing at H, we have 



, sinh 8^ sinh8 c sinh 8 A 



9 = Vz — r - ^ ' • i * • • , „ mhos, (138) 



coshV sinho^, sinh6 B 



and freeing at A, 



. ' • u/> sinh 8/, sinh 8, G /a . . . 



g = Vi sinh^i • . , . • . , . • ~ mhos. (139) 



y J smhbc smb. 8* G fP K ' 



Methods of Computation adapted to Alternating-Current Cases. 



There is especial need for brief methods of computation when 

 A. C cases are dealt with, 6 owing to the complexity of the vector 

 arithmetic. In practice, the degree of precision desired will usually be 

 much lower than that aimed at in the arithmetical examples of this 

 paper, where the numerical values have been carried to six significant 

 digits. Graphical methods may be frequently used with advantage, 

 especially in the vector addition of complex hyperbolic angles. Trav- 

 erse Tables as used by navigators may also be used with advantage 

 for the resolution of vectors into complex quantities. 



The following formulas are also useful: 



cosh (p ±jq) = Vcosh' 2 ^ — sin 2 g/± tan _1 (tanhp-tan</) (140) 



sinh (p ± jq) = yauSFp -f- sin' 2 q /± tan _1 (coth/> • tan q) (141) 



1 ( a. • \ _ sinh 2^? . sin2g , , 



cosh 2 p + cos "2q cosh 2p + cos 2 q 



(148) 



6 A table of hyperbolic tangents of a vector variable or of tanh r/0, is 

 being prepared by the writer for values of r between and 6, by steps of 0.1 

 or less ; and for virtually all angles 0, by steps of one degree. 



tanh 



