82 
1 1 
pe << 5 = ee 9 z 
y’—3cotr [- (a+ 277), 768 2ir) |y 
1 i 1 Z 
+3cotr[ — 3 (A+ 2ir), — 7 (8 —2ir) Jy—1. 
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 formule involving 
tresines, do not, in general, relate to the lengths of lines. 
If x, y, z and a’, y’, 2’ be respectively the rectangular co- 
ordinates of two points in space, the distance between them is 
expressed by the quantity (a — 2)? + (y’— y)? + (2 —z). 
But this is not the function of a, y, z, 2’, y’, 2’ that the for- 
mule in question commonly bring before us: they most fre- 
quently introduce, instead of it, the function 
V (a —2) + (yyy + (2-2)? — 3(a’—2) (y'—y) (= 2) 
which Mr. Graves proposes to call the cubic modulus of the 
line joining the points 2, y, z and 2’, y’, 2’. This eubie mo- 
dulus may be written in a form which suggests important con- 
sequences. 
If we put 
x =meotr[¢,x| a’ = m’cotr[¢’, x’] 
y = mtres[¢, x] y’ = m' tres [9’,y'] 
z= mtres[y,¢] 2’ = m' tres [y’, ¢’] 
it becomes 
s/ m3 —3m”mcotr| p— sx —X'}+ 3m’mcotr[ p’ —$x’-x] —m' 
Thus we have the modulus of the line joining two points in 
space expressed by means of the differences 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 
