a ——— 
FEBRUARY 12, 1914] 
formation and destruction of unstable intermediate 
compounds, the author asserting that ‘this theory, 
in spite of certain imperfections, has been the 
guiding beacon in all his researches on catalysis. 
G. 'T. M. 
MATHEMATICS: PURE AND APPLIED. 
(1) Vectorial Mechanics. By Dr. L. Silberstein. 
Pp. viii+197. (London: Maemillan and Co., 
Ltd., 1913.) Price 7s. 6d. net. 
(2) An Introduction to the Mathematical Theory 
of Attraction. By Dr. F. A. Tarleton. Vol. ii. 
Pp. xi+207. (London: Longmans, Green and 
Co., 1913.) Price 6s. 
(3) A First Course in Projective Geometry. By 
E. Howard Smart. Pp. xxiii+273. (London: 
Macmillan and Co., Ltd., 1913.) Price 7s. 6d. 
(2) R. SILBERSTEIN’S “Vectorial Mech- 
anics” is an able exposition of the 
power of vector analysis in attacking certain types 
of physical problems. Heaviside’s modification of 
Hamilton’s original vector and scalar notations 
is adopted throughout. So far as the simpler 
applications of vector analysis go, the question 
of notation is apparently of little consequence. 
Almost every vector analyst who writes a book on 
the subject has his own pet notation; and there 
is a tendency for these authors to fail to recognise 
that their best creations are usually Hamilton’s 
originals disguised. Even Dr. Silberstein, who 
knows and works quaternions, ascribes to Heavi- 
side a formula given long ago by Hamilton, 
assigns to Clifford (1878) a problem which is 
completely solved in the first edition (1867) of 
Tait’s ‘‘ Quaternions,’’ and refers to Henrici and 
Turner as authorities in connection with a simple 
geometrical problem given in Kelland and Tait’s 
“Introduction to Quaternions.” One might with 
as much historic truth ascribe the proposition 
Euclid i. 47 to the first English examiner who set 
it in an examination paper. Indeed, the historic 
references throughout the book are not all that 
might be desired: For example, it is incorrect to 
speak of Willard Gibbs as the one to whom, after 
Hamilton, the discovery of the fundamental pro- 
perties of the linear vector function is due. What 
of Tait’s powerful paper of 1868 on the rotation 
of a rigid body about a fixed point? It positively 
bristles with new-found. properties and applica- 
tions of the linear vector function. Dr. Silber- 
stein’s own chapter v. is simply a reproduction of 
part of this memoir. Then in the second edition 
(1873) of his treatise on ‘“Quaternions,” Tait for 
the first time develops the application of the linear 
vector function to strains; and in the last chapter 
of Kelland and Tait’s “Introduction to Quatern- 
ions” (1873) presents the theory in a different’ 
NO. 2311, VOL. 92] 
NATURE 
oa Rae Se 
657 
form. Willard Gibbs’s “Vector Analysis” (not 
published) was printed for the use of his students 
in 1881 and 1884... Apart from new names and a 
new and extremely interesting presentation, it is 
doubtful if Gibbs gave in that pamphlet any im- 
portant property of the linear vector function 
which was not to be found in the pages of either 
Hamilton or Tait. 
Then as regards the differential operator V it 
was unquestionably Tait who, first in his paper on 
Green’s and allied theorems (1870), and after- 
wards in his treatise on quaternions (second and 
third editions), developed it and showed forth its 
power. Willard Gibbs got it partly from Tait’s 
“Quaternions ” and partly from Maxwell’s “ Elec- 
tricity and Magnetism”; and Maxwell got it 
directly from Tait. Yet while giving great credit 
to Gibbs and Heaviside, Dr. Silberstein does not 
mention Tait’s name once. The manner in which 
Dr. Silberstein leads up to Stokes’s “Theorem ” 
is not convincing, that is, if the explanation is 
meant to be a proof. Phrases like “we may 
conclude”” and “we may consider” are scarcely 
satisfactory in establishing a far-reaching mathe- 
matical transformation. Moreover, no attempt is 
made to establish the useful vector extensions of 
the theorems of Gauss and Stokes. It is, indeed, 
in these integral theorems involving the vy that, 
as compared with the quaternion vector analysis, 
the artificiality of other vector analyses mainly 
appears. The transformations lack flexibility. 
The reason for this is that outside the quaternion 
vector analysis the reciprocal of a vector is tabu, 
and the associative law in products is despised. 
Apart from the necessary imperfections of a 
non-associative vector algebra, Dr. Silberstein’s 
book contains many good things. In his treat- 
ment of the rotation of a solid body and of strain 
there is not so much of novelty, except when in 
the latter case he considers discontinuous motions. 
In the chapter on hydrodynamics, however, there 
are certain interesting developments which demon- 
strate the directness and value of vector methods. 
On p. 143 the long-winded semi-Cartesian trans- 
formation is needlessly laborious; for at once in 
quaternion notation : 
Soy .¢=VoVyot+V\Soyo=VoVyo+jvo’, 
where o is the fluid velocity. 
' (2) After a lapse of fourteen years Prof. Tarleton 
has brought out the second volume of his “ Intro- 
duction to the Mathematical Theory of Attrac- 
tion,” the first volume of which was reviewed in 
Nature for April 29, 1899. The chapters: are 
numbered consecutively with the chapters of the 
first volume. An elegant discussion of spherical 
and ellipsoidal harmonics occupies chapter viii. 
In chapter ix. the author develops on familiar 
