x Supplement to “ Nature,” March 14, 1907 
himself points out, accompanied by more than an 
added complexity in the movements of this particular 
district. There is a new ‘‘ figure ’’ to equilibrate. It 
might also be suggested that there is something new 
in the location of eyes and semicircular canals at the 
end of so long a flail. 
No one interested in the central nervous system can 
read Prof. Bollx’s book without attention or without 
criticism. J. S. Macponarp. 
PARTIAL DIFFERENTIAL EQUA TIONS. 
Theory of Differential Equations. By Dr. A. R. 
Forsyth, F.R.S. Vol. v., pp. xx+478; vol. vi., pp. 
xiv+596. (Cambridge: University Press, 1906.) 
Price 25s. net. 
HE appearance of these volumes marks the happy 
conclusion of a work undertaken, as the author 
reminds us in his preface, twenty-one years ago. 
Doubtless it would have been finished earlier had 
it not been for unavoidable interruptions; but the 
delay must have brought its compensations, because 
many most interesting developments are of recent 
date. 
Vol. v. deals with equations of the first order, 
and immediately suggests two reflections—one that 
Lie has made the most important contribution to the 
subject since the publication of Jacobi’s memoirs, and 
the other that it is a great help to have such an out- 
line of Lie’s theory, with Mayer’s simplifications, as 
that given in chapter ix. The Jacobian theory, too, 
with Mayer’s developments, is given in chapters iii., 
iv. in a very attractive and readable form. 
f + Chapters 
Vi., Vii., viii. 
are mainly concerned with character- 
istics, and embody much of the work of Cauchy, 
Monge, Lie, and Darboux, as well as original con- 
tributions by Prof. Forsyth himself. 
It may be a rather far-fetched comparison, but 
there does appear to be a kind of analogy between the 
achievements of von Staudt and Lie. Von Staudt’s 
treatise on projective geometry does not contain a 
single diagram, but it is beyond question the most 
masterly work on the subject. Lie is almost, if not 
quite, as chary of graphical illustration, but the spirit 
of his work is geometrical throughout, and he stands 
in the same sort of relation to Monge that von Staudt 
does to Steiner. It is most interesting to see how 
the canonical equations of dynamics (pp. 398-406) 
are illuminated by the theory of contact transform- 
ations; and, again, it is mainly Lie’s ideas which 
have prepared the way for a thorough discussion of 
all the solutions of a partial differential equation, in- 
cluding the special integrals which do not come into 
the ordinary classification. 
The great advance which has been made arises 
from considering a differential equation, not merely 
as representing a property of a function assumed to 
exist, but as defining an aggregate of elements which 
are most vividly realisable in a geometrical form. In 
partial differential equations of the first order these 
elements may be taken to be tiny fragments of planes 
scattered about in space; the differential equation de- 
NQ. 1950, VOL. 75] 
fines the system of elements, and a complete integral, 
if it exist, represents the collecting of the elements 
into surfaces which form a family. In Clebsch’s 
treatise on geometry, there is a chapter on connexes 
to which he evidently attached importance, and 
which has obvious relations, not only to mixed con- 
comitants, but also to ordinary differential equations. 
If it has not been already done, it might be worth 
while to see whether something might not be made 
out of these relations; Clebsch’s work has, of late, 
rather suffered neglect. Again, it may be suggested 
that in dealing with partial differential equations of 
the second order it might be helpful to associate with 
given values (x, y, 2, p, q, 7, Ss, t) a fragment of a 
surface of the second order, just as a fragment of a 
plane is associated with (x, y, 2, p, q). That frag- 
ments of this kind are less likely to be associable so 
as to form surfaces than corresponding plane elements 
is tolerably plain, and partly accounts for the in- 
creasing difficulty of treating equations of the second 
order without making particular assumptions. 
Vol. vi. of the present work is practically devoted 
to partial differential equations of the second order. 
Thus we have chapters on Laplace’s linear equation, 
with the elegant developments of Darboux, Moutard, 
and others; the methods of Monge, Ampére, Boole, 
Darboux, Hamburger, &c., with instructive compari- 
sons, and examples worked out each way; together 
with a chapter on general transformation, embodying 
the most important of Backland’s results. As an 
example of the power of Lie’s methods even in the 
production of beautiful particulary theorems, the pro- 
position on p..295 may be quoted :— 
“When an equation of the second order (of the 
Monge-Ampére form) has two independent inter- 
mediate integrals, it is reducible to the form s=o by 
contact transformations.”’ 
Very little, comparatively, has been done for equa- 
tions of order higher than the second. Prof. Lloyd 
Tanner is one of the few pioneers in this region, 
and his results, obtained by a different method, are 
explained in chapter xxii. 
Prof. Forsyth explains in his nreface and final 
remarks the principles which have guided him in his 
choice of material. This must, indeed, have been a 
most difficult task. It would be easy to dub this 
treatise ‘‘ encyclopaedic,’’ but it is mot, and the fact 
that it is not is one of its merits. The literature on 
ordinary linear equations alone which has been pub- 
lished since Fuchs’s memoir appeared in Crelle’s 
Journal would much more than fill the whole of Prof. 
Forsyth’s pages. No one who is not prepared to 
devote the whole of his time to the subject can 
possibly betome familiar with all that has been 
written about it; and even if, as is quite possible, 
this treatise may occasionally disappoint those who 
consult it on some subsection of the subject in which 
they are specially interested, it is sure to be of great 
service by presenting an ordered and not unwieldy 
body of doctrine, together with suggestions of the 
directions in which further progress may be expected. 
Gay Bae 
