NOVEMBER 19, 1897.] h 
known when those for the zero point are 
assigned; and so on from point to point. 
The trapezoidal rule may be replaced by 
more accurate rules if desired. The method 
is applicable to equations of any order, and 
also to simultaneous equations. 
No. 4 showed that if two matrices are 
commutative, i. e., if ¢4=¢¢, then there 
is no latent system of any root of the one 
which lies in the extension composed of 
two or more latent regions of a root of the 
other unless it includes the entirety of 
these regions. The case when one of the 
matrices has equal roots was developed. 
No.5 presented a new theory of the quad- 
ratie equation x + 2b% + ¢= 0; stating that 
when 0’ is greater than c the roots may be 
either real numbers or ‘hyperbolic com- 
plexes,’ and that when 6’ is less than ¢ the 
roots may be either ‘ circular complexes’ or 
scalar numbers. In this view the square 
root of negative unity can in certain cases 
be interpreted as a scalar instead ofa versor. 
No. 6, which was read in joint session, 
showed by an example that when one term 
in the differential equation is the orthog- 
onal projection of a plane motion, it is in 
some cases easier to pass to the auxiliary mo- 
tion by means of ‘planar algebra’ than it 
is to proceed with the given equation di- 
rectly. (See Trans. A. I. HE. E., Vol. X., p. 
186.) 
In No. 7, of which the Secretary presented 
a brief abstract, there were n— 1 linear 
equations and one quadratic equation in the 
same n variables, and the problem was 
to determine when the simultaneous values 
of the variables that satisfy these equations 
are all real. The criterion obtained has 
applications for n = 2, 3, 4 or 5, whether the 
generating element be a point, a plane, a 
line or a sphere. 
The object of No. 8, which was read by 
the Secretary, was to awaken an interest 
* among astronomers and psychologists such 
as to induce them to pay more attention to 
SCIENCE. 757 
the work of each other and thus improve 
their own methods where necessary. 
In No. 9, which was presented in abstract 
by Dr. Shaw, the idea of obtaining the value 
of A(m) the m™ compound of the determi- 
nant A, as a power of A by multiplying it 
by its adjugate A(n—m), the (n—m)™ 
compound of A, is extended to finding the 
value of certain minors of A(m) in terms of 
Aanditsminors. By making use ofa com- 
prehensive notation the whole subject is 
unified, the laws of vanishing minors set 
forth, and such well-known theorems as 
Sylvester’s and others are easily established. 
In the absence of the author No. 10 was 
read by title, and a printed pamphlet was 
distributed in Sections A and B. 
An abstract of No. 11 was presented by 
the Secretary. It stated that about 70 per 
cent. of the total magnetization of the earth 
can be referred to a homogeneous magneti- 
zation about a diameter inclined to the 
earth’s rotation axis by an angular amount 
of about 12°. This axis has been termed 
by Gauss the earth’s magnetic axis. Itis 
an interesting question to determine the 
motion of this axis during the past two or 
three centuries, and Dr. Bauer’s paper was 
an attempt to solve this problem as far as 
is possible with the data at present at com- 
mand. ; 
No. 12 was also read in brief abstract 
by the Secretary. It stated that as yet no 
formule had been found by which the diur- 
nal range of the magnetic declination, for 
example, could be computed for various 
portions of the earth. The author has 
found the following simple formulze to hold 
true within the fluctuations to which the 
quantities themselves are subject : Diurnal 
range of declination = 2/.58 see’ ¢ ; diurnal 
range of inclination = 6/.1 +1+ 3 sin’ ¢; 
wherein ¢, the magnetic latitude, is found 
from the equation tan ¢g = $tan I, in which 
I is the magnetic dip. The first formula 
was discovered empirically, then under cer- 
