84 
The curve of double curvature, or curve of tresines, defined 
by the equations 
x = cotr[¢, o] 
y = tres [@, 0] 
z = tres[o, ¢] 
appears likewise to be most fertile in properties analogous to 
those of the circle. The following were stated by Mr. Graves: 
1. The angle between the tangents at any two points on 
the curve is equal to that between the two corresponding radii. 
2. The angle between the osculating planes at any two 
points on the curve is equal to the angle between the radii 
drawn to the two reciprocal points. 
3. The angle of contact at a point on the curve is there- 
Sore equal to the torsion at the reciprocal point. 
4. The element of the curve described by the reciprocal 
point is double the elementary area described by the radius 
vector. 
5. The polar reciprocal of the developable surface formed 
by the osculating planes is itself the curve of tresines. 
6. The product of the radii of curvature at any point and 
at its reciprocal is equal to unity. 
7. The radius vector traces a logarithmic spiral upon a 
plane parallel to the symmetric plane. 
Mr. John Neville read a paper on the maximum Amount 
of Resistance required to sustain Banks of Earth and other 
Materials. 
Let CDE be any bank of 
earth, sand, or other material, 
and CE the position of the 
line of repose with respect to 
the horizon and bank ; then if 
we put CA, the perpendicular 
from C on DE produced =f, 
the 2 DEA = 34; the angle 
ECA = ¢; and the 2 ACF, which the fracture CF makes 
