REVIEWS 345 



the approximate solutions so frequently employed by the practical engineer. He 

 proves that in many cases the exact solution can be obtained with less labour than 

 that entailed in using the approximate solutions. 



This book should be read as a sequel to the usual elementary mathematical 

 treatises on the subject — for instance, the introductions to the theory by Fleming 

 or Kennelly. It is not suitable for the student who approaches the problem for 

 the first time. 



In the first two chapters " direct current " lines having leakage are discussed, 

 the solutions being given in terms of hyperbolic functions. They thus form 

 an excellent introduction to the more difficult alternating current case. In 

 Chapter Ilia most excellent discussion, although somewhat difficult to follow, is 

 given of practical harmonic analysis. The author has a thorough grasp of the 

 problem. He contemplates measuring 72 ordinates per wave length, and gives 

 forms to facilitate the calculation. The calculations indicated, however, strike us as 

 being laborious, and, noticing that in practice there are unavoidable errors made, 

 both in drawing the curve and in measuring the lengths of the ordinates, we think 

 that simpler formulae would suffice. It has to be remembered also that an infinite 

 number of curves can be drawn through the extremities of the 72 ordinates, and 

 that the method makes the harmonics of low order have the same value for all 

 these curves — a result which seems improbable. The author recognises this, and 

 says his analysis will give the "simplest" curve drawn through the points. 

 Opinions may differ as to what the simplest curve is. We think mathematicians 

 would say that the curve whose equation was — 



V = a + fl, x + a 2 -r 2 + . . . + a n .r 71 



was the simplest curve, and this certainly is not the author's curve. In our opinion 

 the proper curve must be computed by the theory of probability, and no one has 

 yet done this. 



The concluding chapters give a very thorough discussion of the ordinary 

 differential equations, and are excellently illustrated by diagrams. Many 

 numerical examples are given which prove that when the problem is attacked in 

 the proper way the numerical labour involved need not be excessive. The 

 mathematical appendix given has been taken from the Smithsonian Mathematical 

 Tables, andi is distinctly helpful. Very complete tables of the logarithms of the 

 hyperbolic functions are given, in computing which the author has the help of 

 Mr. B. M. Woods. 



Many novel methods are gi^en which will be helpful in practical work — for 

 example, the methods given of constructing voltage regulation curves. No 

 attempt has been made to discuss how the " skin effect " will modify the solutions 

 given, but perhaps this is reserved for a subsequent volume. We have not noticed 

 any misprints, and the numerical accuracy of the examples is worthy of com- 

 mendation. The want of an index takes away from the Value of the book. The 

 author uses "direct current" and "continuous current," and "arc tan x ,! and 

 "tan -1 .r" indiscriminately. Engineers have now definitely adopted direct 

 current, which is often contracted to d.c. They have long used a.c. for alter- 

 nating current. 



The book can be recommended to mathematicians interested in this important 

 problem, and also to mathematical engineers who have to discuss the long-distance 

 transmission of speech or electric power along wires. 



A. Russell. 



