266 
WATOURE 
[JANUARY 21, 1897 
the means of getting all the results the “spot-lens” can 
give, and we very heartily hope that even low powers 
are now rarely used without a suitable condenser. This, 
however, is a detail, and leaves the instructions to the 
tyro on this head with very little to be desired. 
“The Pond Hunters Museum” and “Aquaria and 
their Management,” are both chapters of great value, 
and they are written by one who has realised the 
pleasures and the difficulties they involve. And after 
this we enter upon the supreme purpose of the book— 
the life which the pond and the stream reveals. For the 
purpose which the writer had in view it is not easy to 
conceive of a more practical and thorough treatment of 
his subject, and withal one which would enable the least 
initiated to follow more intelligently, and at the pond- 
side, what this book incites him to study. 
It is not with the lower and minuter forms of life that 
the author chiefly concerns himself. These are lightly 
touched, affording ample room for future study. But 
worms, leeches, molluscs, crustaceans, spiders, aquatic 
insects, fishes and amphibians, form the main subjects of 
study. 
In this region of study, as in all others, wonderful 
advances have been made. The pond-hunter of twenty- 
five years ago would have found a treasure indeed in a 
book like this. Its thoroughness and its admirable 
illustrations taken together give it a great value to the 
youth who happily determines to make the life of the 
pond or the stream his hobby ; and if it never goes 
beyond that point, this volume will have served an ad- 
mirable purpose. But the book is so well written, and is 
capable of inciting so much interest, that we believe it 
will accomplish a deeper and more lasting purpose 
W., HD: 
THE LUNAR THEORY. 
An Introductory Treatise on the Lunar Theory. By 
Prof. E. W. Brown, M.A. Pp. xvi + 292. (London : 
Cambridge University Press, 1896.) 
HE design of this valuable text-book on the lunar 
theory is similar to that of Tisserand’s “‘ Mécanique 
Céleste,” the object in both cases being to lay before 
the reader the methods by which various practical 
problems of gravitational astronomy have been attacked. 
In each case the recent pure mathematical investigations 
of Poincaré, Lindstedt, Gyldén, &c., though not passed 
by without notice, evidently form but a small part of the 
author’s plan. Of the two writers, Prof. Brown is by far 
the least ambitious ; and his work does not extend, like 
Tisserand’s, to planetary theory, figure of the earth, 
precession, and other gravitational problems that form so 
large a part of the most recent “ Mécanique Céleste.” We 
venture to think, however, that Prof. Brown has dealt 
with his more limited subject in a manner that is far 
clearer, more thorough, and more useful to the student. 
Prof. Brown has not attempted to follow any theory 
through all the approximations that are necessary 
for obtaining an orbit that shall represent the moon’s 
path within the limits of observation, neither are the 
huge masses of figures necessary for such a task repro- 
duced in the treatise. There is no mathematical point 
that cannot be sufficiently illustrated by the third 
approximation, or terms depending on the square of the 
NO. 1421, VOL. 55] 
disturbing force. The author has therefore limited 
himself generally to the first approximation, or inter- 
mediate orbit ; to the second approximation, depending 
on the first power of the disturbing force; and to the 
third approximation, depending on the square of the dis- 
turbing force. In connection with the first approxima- 
tion the author discusses the choice of an intermediate 
orbit, and in the case where this orbit is an ellipse he 
shows why it was necessary to modify it so as to repre- 
sent the motion of the node and apse. Various elliptic 
formula are also given, including the application of 
Bessel’s Functions. A theorem of Hansen’s is also 
given, that is subsequently employed. 
For a second approximation the author shows that in 
practice the earth’s mass may be neglected in comparison 
with the sun’s, and that subject to a simple modification 
in the final result the moon’s mass may be neglected 
altogether, or rather assigned to the earth. A numerical 
estimate—which we believe is original—is given of the 
magnitude of the errors involved in these assumptions. 
In this connection we should like to enter a protest 
against the calculation of terms depending on the square 
of the sun’s parallax when the moon’s mass is neglected. 
The modification, above referred to, does not correct 
these terms; they cannot in any way be made to repre- 
sent an actual phenomenon : they are, as it happens, 
small enough to be negligible—were this not so, the 
method would have to be altered in order to compute 
them. 
The disturbing function is also developed in different 
ways suitable for different theories. The differential 
equations of disturbed motion are also obtained. In the 
integration, various points are carefully discussed. The 
most important of these is the way in which terms pro- 
portional to the time might occur, the way in which such 
terms are got rid of, and the interpretation of this artifice 
—due to Clairaut—as representing a motion of the node 
and apse. Another point is the meaning of the constants 
of integration ; when, for instance, the motion is no longer 
elliptic, the notion of the eccentricity becomes somewhat 
vague. In order to render two theories comparable, the 
arbitrary constants of one must be expressed in terms of 
those of the other; and hence it is desirable in every 
theory to have a clear conception of the meaning—if 
possible the physical meaning—of the constants. Again, 
at every fresh approximation fresh constants arise as part 
of the “complementary function.” What to do with 
these constants requires careful consideration; some- 
times one has to be left arbitrary for a time, in order that 
it may. be used later to remove terms depending on the 
time ; more often—and the preceding case is really only 
a special case of this—they may be used to suitably 
modify or to define more exactly the constants that have 
previously arisen. These points are of fundamenta 
importance, and are rightly dealt with by Prof, Brown at 
considerable length. 
The difficulties of a third approximation chiefly consist 
in the necessity for computing the disturbing forces with 
the disturbed coordinates already obtained. An example 
is given from De Pontécoulant s method. 
Prof. Brown also discusses the general form of the 
final result. Every argument must be the sum or difference 
of integral multiples of four angles, and the characteristic 
