51 
properties of elliptical vibrations as are analogous to those of 
rectilinear vibrations; and it was in this way that the above 
rule was discovered. It is analogous (though it scarcely ap- 
pears so at first sight) to the rule by which, in the theory of 
Fresnel, the direction of rectilinear vibrations is determined, 
when the plane of the wave is given. 
The Rey. Charles Graves read the first part of a paper on 
Algebraic Triplets. 
The object which he proposes to himself is to frame, for 
the geometry of three dimensions, a theory strictly analogous 
to that by which Mr. Warren has succeeded in representing 
the combined lengths and directions of right lines in a plane. 
In carrying out this design Mr. Graves has necessarily been 
led to the consideration of new imaginaries. 
For the sake of clearness it will be desirable to take, in 
the first instance, a brief survey of the fundamental properties 
of algebraic couplets, depending, as they do, upon the nature 
of the symbol /— 1. The correspondence between received 
notions and the views now put forward will thus be made 
more apparent. 
If we take the binomial or couplet x + Y —1. y, in 
which z and y are real quantities, and multiply it by a similar 
couplet a, + a y;, the product will likewise be a bino- 
mial of the same kind, x + / — 1. Y2; and between the 
constituents of the three couplets there exists the relation 
(2? + y?) (a? + 1°) = a? + y.?. (a) 
But couplets may be more readily compared after undergoing 
a simple transformation. Such an expression as 2+ (/—1.y 
may be reduced to the form reV—"- by making r=V/ 2? + 7, 
and = tan-1(2), Hence it appears that if we agree to call 
r the modulus and @ the amplitude of a couplet, the following 
theorems will be true: 
