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included angle. Having now got the notion of the modulus 
of a right line in space, we may easily advance to the con- 
ception of the modulus of a curved line: for we may regard it 
as the sum of the moduli of the elements of the curve: so that 
to find m, the modulus of the curve itself, we employ the formula 
m = § V/ da? + dy? + dx* — 3dx dy dz. 
The right line drawn from the point x, y, z to another point, 
2’, y’, 2’, may be looked upon as having amplitudes as well as 
amodulus. In order to determine them we put 
ax = mcotr[¢, x] :y¥ —y=mtres[ ¢, x]: 2’ —z = mtres[y; ¢]: 
m being the modulus of the right line. 
For the purpose of illustrating his views, Mr. Graves has 
discussed the surface whose equation in rectangular coordinates 
is 
P+ y+2— 3ayz= 1. 
As might be expected, this surface possesses numerous pro- 
perties which admit of being stated in such a manner as to 
exhibit a striking similarity to those of the circle. The three 
coordinates x, y, z belonging to any point on it, may be put 
equal to cotr[¢, x], tres[, x], and tres[y, p]; and we may 
call and y the amplitudes of the point. It will also be con- 
venient to designate the point, whose amplitudes are — ¢ and 
— xX; as reciprocal to the point whose amplitudes are ¢ and y. 
Amongst other theorems relating to this surface, which 
Mr. Graves proposes to name the surface of tresines, the fol- 
lowing may be considered as deserving attention : 
1. The angle between the tangent planes at any two points 
on the surface is equal to that between the radii drawn to the 
two reciprocal points. 
2. The measure of curvature at any point is equal to 
—r,*: r, denoting the radius of the reciprocal point. 
3. The surface is one of revolution : and it is its own polar 
reciprocal with reference to a sphere whose centre is at the 
origin of coordinates and whose radius is unity. 
