107 
yz 
m* =a? y+ 2 + > 
Meee oY hi! Se oar lt (% 
~ = tan (2) : x= tan (=). 
Hence if x, y, z be the rectangular coordinates of a point in 
space, the modulus of the right line drawn to it from the ori- 
gin is a fourth proportional to the projections of this line upon 
the axis of x and the planes of ay and zz. And the amplitudes 
are the angles between the axis of x and these two last pro- 
jections. 
The construction thus obtained for the product of two 
right lines obviously coincides with Mr. Warren’s in the case 
where z = 0. 
yz 
— involves the 
x 
The nullity of the triplet x+iy+jz+i 
three equations, z= 0, y= 0, z= 0. 
But, however well these triplets fulfil the requisitions of 
multiplication, we find they will not stand the test of addition. 
The sum of two such triplets is not necessarily a triplet; nor 
can we add two of them together, unless they happen to have 
_ a common amplitude. 
What has been here said may readily be extended, for we 
might develope the expression e#+Jx-+#¥ + &¢., in which ¢, 7, h, 
&e. represent (n — 1) distinct square roots of negative unity, 
into a series of terms, such as 
A+ iw+jco+Apd+... 
- CD .BD .. BC 
Lk Tidal aki ee ed 
.. BCD 
+ yk-3 + eve 
and, conversely, we may reduce a multiplet 
atibtje+id+... 
