or 
TRANSACTIONS OF THE SECTIONS. 
‘7 —The vertical series are, 
5 
2 
16 
3580 
17956240 
3978175584236200 
19840377976181556439582242163600. 
The next consists of 64 figures, and soon. He pointed out that a similar law ex- 
isted for every other number; and he exhibited formule by which the sum of any of 
the recurrent series may be determined. in the case of 5, S,=2 (S,—1+1), the 
consecutive sums of the several series being 7, 16, 34, 70, 142, &c. In this way 
tables of the powers of numbers may be constructed to any extent whatever with 
very little labour. This discovery will enable certain calculations to be made with a 
degree of accuracy hitherto impossible. 
On the Conditions of Equilibrium in a Rotating Spheroid. 
By Dr. F. A. SinsestROM, of Stockholm. 
In the ‘ Mécanique Céleste’ it is demonstrated that, the earth being considered 
as a fluid mass rotating with a given velocity, the mathematical conditions of equili- 
brium may be satisfied not only by the spheroid of small excentricity, which is the 
earth’s actual form, but also by another quite flat ellipsoidal figure. The following 
remarks may in some measure serve to elucidate this very curious result of the cal- 
culus, and also show that there really is some difference (and that an important ene) 
between the two states of equilibrium corresponding to the said figures. 
Supposing the mass to be in equilibrium with an ellipsoidal figure, the equation 
of condition (though deduced from much more general principles) implies nothing 
more than the hydrostatical equilibrium in the centre of the figure between a polar 
and an equatorial fluid column. _ If, now, the excentricity is thought variable (the 
mass being constant), it can be shown that the value of the equatorial pressure (con- 
sidered as a function of the excentricity) has a maximum, on both sides of which 
there may be a value equal to that of the polar pressure; consequently two states 
of equilibrium and two different figures. 
Considering these two states of equilibrium, it is further shown that if in each case 
the excentricity is thought a little increased or diminished, the corresponding varia- 
tions of the polar and the equatorial pressure will follow in such a manner that 
only with the figure of small excentricity (the earth’s actual form) there will be 
stability of equilibrium. 
Let EBF represent the actual figure of the earth, and sup- 
pose the figure change as DAG. Then the equatorial pressure 
AC will be > polar pressure DC, and consequently there will 
be a tendency with the mass to resume the former figure. But 
if EBF represents the other (elongated) figure of equilibrium, 
and the excentricity is likewise thought a little increased, then 
the pressure of AC will prove < pressure of DC, and con- 
sequently the mass will have a tendency still more to change 
in the same direction. 
As for Jacobi’s famous ellipsoid of three different axes, it will in no°way, if ex- 
4 amined in this manner, afford stability of equilibrium. Consequently with a given 
velocity of rotation there can be only one ellipsoidal figure with which a fluid mass, . 
subject to its own attractive forces, may remain in a real state of equilibrium. ; 
On a Mode of constructing the Rectangular Hyperbola by Points. 
; By G. Taurnect. 
This communication was illustrated by two figures. In the first was shown how 
by means of lines drawn in an isosceles triangle parallel to its base, on which were 
formed concentric arcs, and through which was drawn, parallel to the base, a sectiov 
