CHAPTER VI. 



CONVERGENT SERIES. 



THE mathematical expression for finding the operating 

 characteristics of a . transmission line, in which exact ac- 

 count is taken of all the electrical properties of the line, 

 has been published many times. It involves the use of 

 hyperbolic sines and cosines, as well as of complex quan- 

 tities,* and, without some special arrangement, cannot be 

 directly applied to the calculation of a particular case. 

 For this reason, most of the systems so far published for 

 calculating transmission lines have used approximate for- 

 mulas which have been based on the hyperbolic formulas. 

 In a few cases, an attempt has been made to devise a 

 system o>f working which would give the exact results of the 

 fundamental hyperbolic formulas, but generally the labor 

 required in using the systems is so great as almost to pro- 

 hibit obtaining the exact result, or else the accuracy of 

 the work is seriously impaired by the necessity of inter- 

 polating values from tables of hyperbolic functions which 

 have been recently prepared for this purpose and are not 

 as large and complete as they should be for good working. 



The original hyperbolic formulas can be expressed in 

 the form of convergent series. f In this form they do not 



* The hyperbolic formulas are given in Chap. XV. 



t Prof. T. R. Rosebrugh, Applied Science Magazine, University of To- 

 ronto, March, 1009; Prof. T. R. Rosebrugh, Proc. A. I.E. E., Nov., 1909, 

 p. 1460; J. F. H. Douglas, Electrical World, April 28, 1910; Dr. C. P. 

 Steinmetz, Electrical World, June 23, 1910; Dr. C. P. Steinmetz, "Engi- 

 neering Mathematics," Chap. V, 1911. 



41 



