82 

 ^,3_3cotr[l(A + 2^V), i(B-2eV)]z/2 



+ 3cotv[-^(^ + 2ir), -^(B-2iT)]t/~l. 



From the result just obtained Mr. Graves deduces a sin- 

 gular geometrical theorem analogous to the celebrated one of 

 Cotes. 



But it must be observed here that the geometrical propo- 

 sitions to which we are led in interpreting formulae involving 

 tresines, do not, in general, relate to the lengths of lines. 

 If X, y, z and x', y', z' be respectively the rectangular co- 

 ordinates of two points in space, the distance between them is 

 expressed by the quantity V {x' — a;)^ + {y'— yY + {z' — ^)^ 

 But this is not the function of x, y, z, x', y', z' that the for- 

 mulae in question commonly bring before us : they most fre- 

 quently introduce, instead of it, the function 



:j(x'-xf + {y>-yf -f- {z'- zY - Z{x'-x) {y'-y) {s,'~z) 

 which Mr. Graves proposes to call the cubic modulus of the 

 line joining the points x, y, z and x', y', z' . This cubic mo- 

 dulus may be written in a form which suggests important con- 

 sequences. 

 If we put 



a; = m cotr [^, -^ x' = m'cotr [0', -^'1 



y z= m tres [^, ^] y' = m! tres [^', y['\ 



zzz. m tres [x, <f\ z' = m' tres [;^', ^'] 



it becomes 



Thus we have the modulus of the line joining two points in 

 space expressed by means of the diiFerences of their corres- 

 ponding amplitudes and the moduli of the right lines drawn to 

 them from the origin : — a result analogous to the fundamental 

 proposition in plane trigonometry, by which the length of one 

 side of a triangle is found from the two remaining sides and the 



