756 
12. Simple expressions for the Diurnal 
range of the Magnetic Declination and of 
the Magnetic Inclination. By Dr. L. A. 
Bauer, University of Cincinnati, Cincinnati, 
Ohio. 
13. The Theory of Perturbations and 
Lie’s Theory of Contact-transformations. 
By Dr. E. O. Lovett, Princeton, N. J. 
14. On Rational Right Triangles. No. 
I. By Dr. Artemas Martin, .U. 8. Coast 
Survey, Washington, D. C. 
15. Some Results in Integration expressed 
by the Elliptic Integrals. By Professor 
James McMahon, Cornell University, Ithaca, 
Ilo Wo 
16. Modification of the Hulerian Cycle 
due to Inequality of the Equatorial Mo- 
ments of Inertia of the Earth. By Profes- 
sor R. 8. Woodward, Columbia University, 
New York. 
17. Integration of the Equations of Ro- 
tation of a Non-rigid Mass for the case of 
Hqual Principal Moments of Inertia. By 
Professor R. 8. Woodward, Columbia Uni- 
versity, New York. 
’ 18. General Theorems concerning a cer- 
tain class of Functions deduced from the 
properties of the Newtonian Potential Func- 
tion. By Dr. J. W. Glover, Ann Arbor, 
Mich. 
19. The Importance of adopting Stand- 
ard Systems of Notation and Coordinates 
in Mathematics and Physics. By Professor 
Frank H. Bigelow, U. S. Weather Bureau, 
Washington, D. D. 
20. A Remarkable Complete Quadrilat- 
eral among the Pascal Lines of an Inscribed 
Six-point of a Conic. By Professor R. D. 
Bohannan, Columbus, Ohio. 
21. Stereoscopic Views of Spherical Cate- 
naries and Gyroscopic Curves. By Profes- 
sor A. G. Greenhill, Royal Artillery Col- 
lege, Woolwich. 
No. 1 pointed out that to every simple 
isomorphism of a group to itself corre- 
sponds some substitution of its operators ; 
SCIENCE. 
[N.S. Vou. VI. No. 151. 
and that to all such isomorphisms corre- 
sponds asubstitution group, which has been 
called the group of isomorphisms of the given 
group. A new and simple proof was given 
of the following theorem of Jordan’s: 
When a regular group (R) of order n is 
transformed into itself by the largest possi- 
ble group (L) of its own degree, the sub- 
group of L which includes all its substitu- 
tions that do not contain a given element is 
the group of isomorphisms of R. Other 
theorems were proved regarding those iso- 
morphisms which may be derived from a 
given one by means of real transforming 
Operators. 
In No. 2, of which a brief abstract was 
read by the Secretary, the general group is 
that of Lie’s Kugelgeometrie. All infinity 
in space is regarded as a single point and 
all planes as spheres through the point at 
infinity. The general group is tenfold. 
All transformations leaving a point invari- 
ant form a sevenfold sub-group. There is 
a sixfold subgroup which leaves a sphere 
invariant, and which is identical with the 
sixfold group of circular transformations on 
the Neumann sphere or in the complex 
plane. 
No. 3 was read in joint session of Sections 
A and B. It showed how to compute suc- 
cessive values of a function defined by a 
differential equation without solving the 
equation analytically. Using Newton’s 
notation for derivatives of y as to x, and 
indicating successive values of y by sub- 
scripts, the simple trapezoidal rule would 
serve to express approximately the differ- 
ence between y, and y, in terms of y, and 
us and similarly that between 7, and y, in 
terms of y, and y,. Hence y, and y, may 
be ultimately expressed in terms of y,, %) 
Yo Jy Substituting their values in the dif- 
ferential eqnation, the latter will give the 
value of y, This value substituted back 
will give values of 7, and y,, and hence the 
quantities for the point 1 are completely 
