599 
‘NATURE 
[OcToBER 19, 1899 
Mr. Heaviside’s treatment of Bessel’s functions is in- 
teresting and suggestive, but the lack of formal arrange- 
ment is here severely felt; it is not always easy to 
distinguish clearly between what is proved and what is 
experimentally assumed to see how it goes as Mr. 
Heaviside puts it. The student who is previously un- 
acquainted with the properties of these functions will 
probably find difficulty in following some of the equations 
written down without proof. In the equation for Ky (gx), 
p. 226, for example, all the information given about y 
(Euler’s constant, but not distinguished as such) is 
‘““where y = 0°5772 is a certain constant introduced to 
make Ky (g¥) vanish at infinity”; certain of the con- 
jugate relations are also without proof, but these possibly 
are left as exercises for the student. 
The work is nevertheless one which will well repay 
careful attention.’ As has been remarked by Prof. De 
Morgan : 
“The history of algebra shows us that nothing is more 
unsound than the rejection of any method which naturally 
arises, on account of one or more apparently valid cases 
in which such methods lead to erroneous results. Such 
cases should indeed teach caution, but not rejection.” 
Mr. Heaviside is much to be congratulated on the 
light he has thrown on difficult and perplexing questions 
in both physics and mathematics, and also for calling the 
attention of mathematicians to a powerful, but somewhat 
neglected, weapon. C. S. WHITEHEAD. 
OUR BOOK SHELF. 
Catalogue of the Lepidoptera of Northumberland, Durham 
and Newcastle-upon-Iyne. Part 1. By J. E. Robson. 
Nat. Hist. Trans. of Northumberland, Durham and 
Newcastle-upon-Tyne, Vol. xii. Part 1. Pp. 195. 
THE present instalment of this important catalogue in- 
cludes the butterflies, together with such of the moths as 
are comprised in the Spingina (hawk-moths), Bom- 
bycina, and WVoctuina. In his classification the author 
thus far follows Mr. Barrett's monograph of the British 
Lepidoptera, to the unpublished portions of which he 
has been supplied with references by Mr. Barrett himself. 
Whatever faults there may be in the scheme of classifica- 
tion in question, and the nomenclature employed therein, 
the adoption of a uniform system by different writers is 
highly desirable ; and we, therefore, consider that Mr. 
Robson has been well advised in the course he has 
adopted. 
As the author has had the advantage of the co- 
operation of all the local collectors of repute, his work 
may be regarded as a thoroughly up-to-date account of 
the Lepidopterous fauna of the northernmost counties of 
England. And how different this fauna is from that of 
the midland and southern counties may be gathered from 
a glance at the portion devoted to the butterflies. The 
common Brimstone Butterfly, for example, is only known 
in the area treated of by two or three stragglers, its 
normal range not extending northwards of South York- 
shire. Much more remarkable, however, is the circum- 
stance that certain species of butterflies, such as the 
Comma and the Red Admiral, which were once common 
in the two counties, have for the last forty years been 
extremely scarce, although the second of the two men- 
tioned has once again become a familiar object since 
1893. It would be interesting to know the reason why 
so many of these insects left the district during the 
NO. 1564, VOL. 60] 
sixties ; but on this point the author is silent. On the 
other hand, as might perhaps have been expected, 
migratory species, such as the Clouded Yellow and the 
Camberwell Beauty, which visit England at uncertain 
intervals in larger or smaller numbers, commonly travel 
into the northern counties ; the author remarking of the 
last-named insect that it “visits these counties on most 
of those rare occasions when a wandering horde strikes 
our shores.” Of the moths, it must suffice to say that the 
Death’s-head has occurred in both counties, and there is 
reason to believe has bred in them, but that the stock is 
probably maintained by immigration from the south. 
The foregoing instances demonstrate that Mr. Robson’s 
work is very far from being a mere dry catalogue ; and 
that it really teems with interesting observations on the 
life-history and distribution of all the species recorded. 
If the sequel be maintained at the same high level, the 
complete catalogue ought to prove a very important con- 
tribution to entomological literature. R. 
The Process Year-Book for 1899 (‘ Penrose’s Pictorial 
Annual”). Edited by William Gamble. Pp. viii + 108. 
(London: Penrose and Co., 1899.) 
THISs is the fifth year’s issue of this most excellent review 
of the graphic arts, and the editor, together with all his 
co-workers, are to be congratulated on the production of 
such a handsome and interesting volume. 
As in former years, most of the articles are written by 
those who are at work in some line of process work, and 
as these are by no means few in number, the reader is 
made acquainted with a great amount of experience 
which may help him to success in the future. The 
feature of the book is undoubtedly the beautiful illus- 
trations, which bring home to the reader the high state 
of excellence that the art of reproduction has reached at 
the present day. All kinds of subjects, from a stellar 
cluster down to an orchid, are illustrated, and these serve 
as types for showing the results obtained by the working 
of different processes. 
The high order of merit attained should not only 
render the book a valuable aid to the process worker 
and others interested in the art of reproduction, but 
should find many other friends who would delight to 
possess such a charming collection of high-class illus- 
trations. 
Mathematical Tables. 
W. Haldane Gee. 
and Co., Ltd., 1899.) 
THIS set of useful tables ina compact form are abstracted 
from the compilers’ larger volume of “ Mathematical and 
Physical Tables.” The idea of this present issne is to 
place before students tables which are suitable for the 
class and laboratory, and which give sufficient accuracy 
for such computations. ' 
To sum up the contents, we have four place logarithms 
and antilogarithms, natural sines, cosines, and tangents, 
with interpolation to 1’. Logarithmic sines, cosines, and 
tangents with differences also to 1’. Tables of squares, 
exponential functions, weights and measures, and finally 
a table of conversion for the last mentioned. 
By James P. Wrapson and W. 
Pp.. 28. (London: Macmillan 
Opinions et Curiosites touchant la Mathématique. By 
G. Maupin. (Paris: Carré et Naud, 1898.) 
THis is a collection of curious ideas and essays, which 
the author has encountered in the course of much 
heterogeneous reading in ancient scientific works, in 
which there has been found any reference however 
remote to mathematical thought. Paradoxes and ab- 
surdities alone seem to be considered worth inclusion ; 
the book is of little or no use as a contribution to the 
history of mathematics. 
