690 
The elements of the theory of surfaces are in- 
troduced in the eighth chapter and applied to 
ruled surfaces and envelopes. The theory of 
curvature is elaborated in detail, including the 
notions of mean curvature due to Germain, 
total curvature conceived by Gauss, and quad- 
ratic curvature of Casorati. The chapter con- 
cludes with the determination and properties of 
the remarkable lines of a surface. 
The ninth chapter begins the study of the in- 
verse problems by presenting the fundamental 
concepts and rules of simple and multiple inte- 
gration. The tenth chapter evaluates the well- 
known forms of rational, irrational and trans- 
cendental indefinite integrals, and terminates 
with certain classes of definite integrals, includ- 
ing elliptic and eulerian integrals; the nature 
of thee xample is indicated by the following, 
which occurs in the study of vortices: 
ab 20 
Qn 
cos Ad0 
V a2 + b? + c2— 2ab cos 4 
After making the ordinary applications to 
mensuration in the eleventh chapter, the author 
undertakes the elements of the theory of differ- 
ential equations in the twelfth chapter. The dis- 
tinctions between the notions general, particu- 
lar, and singular integral are clearly made. The 
cases of integrable ordinary differential equa- 
tions are classified as follows : 1° variables sepa- 
rable ; 2° functions homogeneous ; 3° one varia- 
ble absent ; 4° second order equation lacking one 
variable always reducible to one of first order ; 
5° linear equation; 6° Bernouilli’s equation ; 
7° Clairaut’s equation ; 8° the form y= x¢ (y’) 
+ (y’), when not a Clairaut equation is reduci- 
ble to a linear equation ; 9° Riccati’s equation 
and its characteristic property that the anhar- 
monic ratio of any four particular integrals is 
constant.‘ No reference is made to Lie’s theo- 
ries. A well selected list of resolved problems 
is followed by geometrical applications of dif- 
ferential equations to plane curves, trajectories 
and surfaces. The general linear equation and 
equations with constant coefficients are studied 
somewhat in extenso. Passing then to equations 
in more than two variables, the author takes up 
total differential equations and simultaneous 
ordinary equations and terminates the chapter 
SCIENCE. 
[N. S. Von. X. No. 254. 
with a short treatment of the partial differential 
equation. 
The last chapter of the book gives the ele- 
mentary notions of the calculus of variations in 
six pages. The volume concludes with ‘notes 
on the concept of limit, oscillatory extremes, 
demonstration of Cantor’s theorem, Hadamard’s 
theorem, minima and maxima of functions, 
cusps and flexions at a pole, torsion of curves, 
calculation of the curvature of a surface, 
formule of Rodrigues, general formula of 
Stirling. 
E. O. LOvETT. 
PRINCETON, NEW JERSEY. 
Pflanzen- und Tierverbreitung, in Hann, Hochstet- 
ter und Pokomy, Algemeine Erdkunde. By A. 
KrrcHHorr. Verlag, F. Tempsky, Wien. 
Aufl. 5. 1899. 
This volume, by Alfred Kirchhoff, forms the 
third part of the new edition of a well-known 
and compendious manual of pure as distin- 
guished from economic geography. It main- 
tains the high standard of the preceding parts 
by Hann and by Brickner, and is a welcome ad- 
dition to the literature of geo-biology. Of the 
157 figures, a large proportion are not easily 
accessible elsewhere or are quite new. The 
maps, while not emphasizing the developmental 
phases of faunal and floral distribution as do, 
for example, those of Engler, are, neverthe- 
less, more nearly in accord with modern ideas. 
than those of Grisebach or Decandolle. The 
ecological factors are, by no means, neglected, 
as they were so generally in the older books. 
While it is true that they are scarcely so ex- 
haustively discussed and laboriously analyzed 
as in the special treatises of Warming and 
Schimper, yet they are clearly, ably and ade- 
quately presented. Kirchhoff’s work, has a 
certain advantage over the special Tierlebens and 
Pflanzen-geographies in its broad outlook upon 
both the fields of biological science. It falls 
naturally enough into three divisions, the first 
including the general discussion of the rela- 
tions between the earth and the organisms that 
inhabit it, the second comprising the analysis of 
floral, and the third that of faunal regions. The 
peculiar excellence of the treatment is apparent 
at once in the opening chapters on the migrations. 
of organisms, on the environmental conditions 
