128 The N.Z. Journal of Science and Technology. [Mar. 
to the American Institute of Electrical Engineers.* * * § These also entail a 
considerable amount of interpolation, resulting in delays and uncertain¬ 
ties, and more detailed tables are required. 
The work of calculation would be very much facilitated if more 
detailed tables of the hyperbolic logarithms, the hyperbolic sine and 
cosine, were available, but no advance has been made in this direction 
for twenty-five years. New tables have appeared from time to time, but 
no improvement has been effected in the direction of calculating inter¬ 
mediate values. The object of the compilers seems to be to assemble as 
many samples of tables of different function as possible. 
In the present paper the object of economizing time and facilitating 
calculations is attained by the preparation of a series of detailed tables 
limited in their application to those sizes of conductors generally 
employed for transmission-line work. Engineers soon realize that in 
their particular practice certain sizes of wire are used to the exclusion of 
others, and, recognizing this, the functions involved have been calculated 
for certain cable sizes in standard wire-gauge and for spacings usually 
employed in practice. 
The method of treatment adopted is that employed by Oliver 
Heavyside and Professor Kennedy—the method of vector algebra— 
because it lends itself to easy manipulation and ready evaluation. The 
best single treatise on vector algebra and trigonometry is Hayward’s,! 
but at the same time the chapters devoted to the subject in Chrystal’s 
Algebra should not be overlooked, whilst the best treatise on the appli¬ 
cation of hyperbolic functions to engineering problems is Kennedy's.! 
Several other works may be consulted with advantage, of which a list is 
given in Kennedy’s book above referred to, including his own works. 
These should be studied by those wishing to master the subject, 
because of the lucidity of the treatment and the wealth of example 
and application of the theory to ad the problems of transmission, 
whether in the telegraph or telephone or power circuit. 
Characteristic Equations. 
If v be the instantaneous potential difference and c the current at 
distance x from the end of a transmission circuit, R the resistance, L the 
inductance, K the leakage conductance, S the capacity (or “ permittance,” 
as Heavyside§ prefers it) per unit length of the circuit, we have the 
following equation of connection between the quantities :— 
civ 
dx 
R + L 
d 
dt 
c and - fk = (K+S — 
dt \ dt 
v. 
If v varies with the real part of e^ 1 the operator ip may be substituted 
for d/dt where i = V — 1; and where p = % rf is the angular speed in 
* A. E. Kennelly, Tallies of Hyperbolic Functions in reference to Long 
Alternating-current Transmission-lines, I'roc. Am. Inst. Elect. Engineers , vol. 30, 
pt. 3, pp. 2495-2506, 1911. 
t R. B. Hayward, M.A., F.R.S., The Algebra of Coplanar Vectors and Trigo¬ 
nometry. Macmillan and Co., 1892. 
[A. E. Kennelly, M.A., D.Sc., The Application of Hyperbolic Functions to 
Electrical Engineering Problems. University of London Press, 1912. 
§ Oliver Heavyside, Electrical Papers , vol. 2, p. 125. Macmillan and Co., 
London, 1892. 
