83 



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 = S \/ dx^ -{■ dy'-^ + dx-^ —'idxdy dz. 

 The right line drawn from the point x, y, z to another point, 

 z', y', z', may be looked upon as having amplitudes as well as 

 a modulus. In order to determine them we put 

 od—x - m cotr[(^, x] :z/'-y=W2tres[(^, x] : z'-2; = mtres[x,0]: 



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 



x^ + y^ + z^ — ^ xyz — 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]» t^^s [0, x]? and tres [x, ^] ; and we may 

 call ^ and x the amplitudes of the point. It will also be con- 

 venient to designate the point, whose amplitudes are — ^ and 



— Xj as reciprocal to the point whose amplitudes are ^ and x- 



Amongst other theorems relating to this surface, which 

 Mr. Graves proposes to name the surface oftresinesy 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 



— ^i~* : fi 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. 



