84 



The curve of double curvature, or curve oftresines, defined 



by the equations 



X = cotr [^5 o] 



y = tres [0, o] 



z = tres [o, 0] 

 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- 

 fore 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, / ^ ^.^e'^'/ 



and CE the position of the ^^^^^0^^/ 



line of repose with respect to a ,--'''' '^^^^pT.v^'H 

 the horizon and bank ; then if p. ,^\ ^^ BW:/ \ 

 we put C A, the perpendicular \ y^^' '^\ /' ...--'" 



from C on DE produced = A, \;^^^-"' ° 



the Z DEA — b\ the angle ---^^-^h^^ss^'- 



EGA ■=. c; and the Z ACF, which the fracture CF makes 



