410 
NALORLE 
[AucusT 31, 1899 
THE NEWTONIAN POTENTIAL. 
Théorie du Potentiel Newtonien. By H. Poincaré. Pp. 
366. (Paris: Georges Carré and C. Naud, 1899.) 
HE course of lectures given by Prof. Poincaré at the 
Sorbonne during the session of 1894-5 has, under 
the editorship of Dr. Edouard Leroy and M. Georges 
Vincent, assumed the form of a text-book on attractions 
and the theory of the potential. 
The subject-matter naturally falls into two sections, 
one referring to special properties of potentials of linear, 
superficial and volume distributions, and the other deal- 
ing with Dirichlet’s problem and its solution. It is rather 
a pity that this division was not adhered to in the 
arrangement of the text. Chapter vi., dealing with the 
potentials of magnetic shells, is quite out of place in the 
middle of Dirichlet’s problem, and should logically have 
preceded the two previous chapters. 
In opening what we have regarded as the first subject, 
M. Poincaré introduces concurrently with the Newtonian 
potential the logarithmic potential corresponding to the 
law of the inverse distance, which represents the two- 
dimensional potential of infinite cylindric distributions. 
The first chapter, which includes calculations of the 
potentials of rods, cylinders, spheres, and other simple 
forms, deals with potentials of bodies at external points. 
It contains a brief account of Legendre’s coefficients. In 
passing to the interior of the attracting mass in Chapter ii., 
the question of the convergency of the integrals repre- 
senting the potential and its derivatives naturally 
necessitates a brief digression on convergent integrals in 
general. Chapter iii. deals with potentials of linear and 
superficial distributions of matter, and naturally leads on 
to the misplaced Chapter vi., which treats at considerable 
length of “double layers” (douwdles couches)—in other 
words, magnetic shells. 
The second subject opens in Chapter v., where 
Dirichlet’s problem is stated, the principal properties of 
Green’s function are proved, and the equivalence of the 
two problems is established. In the next chapter Prof. 
Poincaré gives the solutions of Dirichlet’s problem for a 
circle and a sphere, and deals with the properties of con- 
jugate functions and conformal representation in two 
dimensions. Chapter vii. treats of the method of ex- 
haustion (éa/ayage), and the remaining eighty pages 
contain a fairly detailed account of Neumann’s method 
and its extensions. 
Lecture notes are rather apt to be deficient in explan- 
ation on points which have either been taken for granted 
by those who transcribed them, or have been incidentally 
explained in a conversational way by the lecturer. Any 
one not starting with a previous knowledge of the 
definition of the potential would hardly find M. Poincaré’s 
opening very clear. In first “letting” /,'(7,) be the 
attraction at distance 7, and afterwards defining the 
potential as — 3/(7) it ought to be explicitly stated that 
/\'(7,) is the derived function of the subsequently intro- 
duced function f(7,). Moreover, why should the con- 
stant of integration in /,(7,) be taken as zero in the 
Newtonian and as —# log 7) in the logarithmic poten- 
tal? A few additional words of explanation in such 
cases would often save readers from wasting time over 
unnecessary difficulties. 
NO. 1557, VOL. 60] 
There are many problems which, although belonging 
to the subject proper of attractions and potential, are not 
included in the present volume. The potentials of 
ellipsoids are untouched, Lame’s ellipsoidal harmonics 
being dismissed with a mere reference. Then, again 
more might have been said about spherical harmonics. It 
will be seen, however, that M. Poincaré’s lectures have 
reference to the general theory of the potential rather 
than to special problems, which find appropriate treat- 
ment elsewhere. 
As an introduction to this theory dealing at some 
length with Dirichlet’s and Neumann’s developments, 
M. Poincaré’s volume bids fair to be a useful addition 
to the library of college lecturers as well as of the more 
advanced class of mathematical students. Ga HB: 
OUR BOOK SHELF. 
Faune de France—Mammuyéres. By A. Aclogue. 
Pp. 84; Figs. 9. (Paris: Bailliére.) 
As compared with that of the British Isles, the mam- 
malian fauna of France is much more extensive, com- 
prising a number of Mediterranean types quite unknown 
among the former. It is therefore, altogether apart from 
patriotic considerations, well worthy of being separately 
monographed. This task kas been undertaken by the 
author of the present little volume ; and although in the 
main the very condensed descriptions given appear 
satisfactory so far as they go, we cannot but regret that 
the work was not written’ more on the lines of Bell’s 
“British Quadrupeds.” 
The volume commences with an illustrated dissertation 
on the characteristics of, first, the Vertebrata and then of 
mammals ; and in this part we notice that on p. 21 the 
author figures the skull of a bat as that of a mole, and 
also one of a porcupine as that of a second representative 
of the insectivorous order. 
The illustrations are, indeed, very discreditable, the 
only passable ones being those borrowed from other 
works. In these days of cheap ‘“process-blocks” it 
does seem inexcusable to issue caricatures like those 
in the present volume. The type, too, is extremely 
small. 
The descriptions of the genera and species, although, 
as already said, very short, are sufficient to admit of their 
identification. Some of the terms used, such as (p. 73) 
“ Bosidi”—the equivalent of Bovzdae—sound, however, 
somewhat strange to English ears ; and it may be added 
that the nomenclature generally is by no means altogether 
up to date. Moreover, even if it be considered advisable 
in a work of this nature to introduce the ordinary 
indigenous domesticated animals, such as sheep and 
oxen, there seems little to justify the inclusion of such a 
palpable foreigner as the guinea-pig. 
The best we can say is to express the hope that the 
author may, before long, see his way to reissue what 
forms the rudiments of a very useful work on a scale 
more commensurate with the importance and interest of 
the subject. 
Anatomical Diagrams for the use of Art Students. By 
James M. Dunlop. Pp. 72. (London: George Bell 
and Sons, 1899.) 
As to how much or how little knowledge of anatomy the 
art student should possess is a matter on which opinion 
is very much divided. Your youthful impressionist is apt 
to sneer at anatomy ; as arule, his contempt for the sub- 
ject is revealed in the construction of the forms he repre- 
sents. On the other hand, the more serious-minded and 
studious of the artistic fraternity, those who, by hard 
work and diligent study, are laying the foundations upon 
