NATURE 
589 
THURSDAY, OCTOBER 109, 1899. 
ELECTRO-MAGNETIC THEORY. 
Electvo-magnetic Theory. Vol. ii. By Oliver Heaviside, 
Pp. xvi + 542. (London: The Electrician Co., Ltd.) 
HIS interesting work, the first volume of which 
appeared some five years ago, well sustains Mr. 
Heaviside’s reputation as an original investigator, and 
even when we do not agree with his procedure, we must 
admire his fertility of resource and the skilful manner 
in which he develops his methods. Although we are 
more than once warned that the treatment is not formally 
or logically arranged, as is indeed the case, Mr. Heaviside 
has nevertheless, in essentials, admirably arranged his 
matter, so that we are led on by gentle steps from 
comparatively simple to more complex problems. 
The book may be regarded from two distinct points 
of view. Firstly, without inquiring too closely into the 
validity of the mathematical methods employed, we may 
consider the work from a physical point of view as a 
mathematical theory of the propagation of plane electro- 
magnetic waves in conducting dielectrics, according to 
Maxwell’s theory, oras the theory of the propagation of 
waves along wires. Secondly, we may consider the 
book from a purely mathematical point of view as an 
introduction to the theory of generalised differentiation, 
divergent series, and Bessel’s functions, viewed, however, 
for the most part through physical spectacles. 
The book opens with a discussion of the age of the 
earth, in which Prof. Perry’s results are explained and 
contrasted with those of Lord Kelvin. Then follows a 
discussion of the equations 
where V and C are the voltage and current, R and S the 
resistance and permittivity per unit of length, and / stands 
d 
dt 
some detail, and it is very noticeable how easily terminal 
conditions are dealt with by Mr. Heaviside’s methods, 
and in this respect they have a great advantage over 
Fouriers method. The more general equations 
dV _ 
Be =(R+L¢)C, 
for A large number of problems are considered in 
2G 2 
Te SAN 
where L is the inductance and K the leakance, as Mr. 
‘Heaviside terms it, per unit of length, are next con- 
sidered. These in the case where 1/R, 1/L, K and S all 
vary as the 7th power of the distance from x=o lead to 
Bessel’s functions. As before, a great variety of interesting 
and important questions are dealt with, and Mr. Heavi- 
side is careful to explain that these are not mere mathe- 
matical exercises, but that the formule apply to 
cylindrical electro-magnetic waves. The case of R, L, 
K, S, constants is discussed at some length, and owing 
to the application of the results to practical questions 
concerning telegraph and telephone cables they should 
be kept in mind by “practicians.” Mr. Heaviside has 
for long been preaching in the wilderness on this matter, 
but his labours will bear fruit one day, and we trust that 
when the day comes it will not be a case of “tulit alter 
honores,” as has happened to other men in other matters. 
NO. 1564, VOL. 60] 
Some sections are devoted to discussions of the experi- 
ments of Dr. Barton and Dr. Bryan, of spherical waves, 
and, with some reserve, to the experiments of Hertz and 
Lodge. The sections on spherical waves have, as is 
pointed out by the author, a practical application in wire- 
less telegraphy. Some rough, but interesting, curves 
showing the progress of a wave under various circum- 
stances conclude the physical portion of the work. 
Passing on to the mathematical aspect of the book, 
operational methods are freely employed, and their 
reduction to algebraical form leads us at an early stage 
of the work to the question of fractional differentiation. 
This is a subject which has frequently occupied the 
attention of mathematicians, and two main modes of pro- 
ceeding have been proposed, one taking e”, the other x” as 
the fundamental symbol ; the first method was employed 
by Liouville and Kelland, the second by Peacock. 
Both methods find formule: which are certainly true 
when the index of the operating symbol is an integer, 
and for the case of the index or fraction both appeal to 
the principle of the permanence of algebraical forms. If 
both methods produced the same result in every case all 
might be well, but most unfortunately this is not so, at 
least without some further assumption, and it is a ques- 
tion beset with difficulties which system, if either, is to 
be considered the true one. Mr. Heaviside’s method 
evades rather than elucidates the difficulties. He re- 
quires to find the value of #1, where p = f,and tis that 
é 
function of ¢ which is zero before and unity after =o. 
To effect this he takes a suitable physical problem, and, 
solving symbolically, obtains a solution involving !1 ; 
then by another method he finds a solution free from 
operators ; a comparison of the two gives f/1=(m?)™'. 
This is the same value as is given by Peacock’s method, 
but not that which is given by Liouville’s and Kelland’s 
without further assumption. In Chapter vil. another 
way, on the same lines as before, is given of finding this 
result, and the remark is added, “I do not give any 
formal proof that all ways properly followed must neces- 
sarily lead to the same result.” It is much to be re- 
gretted that no hint is -given on this point, for, granting 
that there is a theory of fractional differentiation, the way 
to be properly followed is the essence of the whole 
matter. 
Some of Mr. Heaviside’s methods of dealing with 
series in Chapter vill. are also open to some objection ; 
he more than once tests the equivalence of two series by 
giving the variable numerical values and seeing if the two 
series give the same result. This may be an “excursion 
to the borders of the realms of duplicity,” but scarcely to 
those of “fearful rigour.” It would seem, indeed, from 
many passages in the book, that Mr. Heaviside considers 
rigour in mathematics to be of somewhat minor im- 
portance ; for instance : 
“You have first to find out what there is to find 
out. How you do it is quite a secondary consideration.” 
If this advice were to be generally followed, mathe- 
maticians would no doubt jump many gates in their 
endeavours to reach the goal on the other side, but 
whether or no they would not at times land in a quag- 
mire may be open to doubt. 
iE 
