51 



properties of elliptical vibrations as are analogous to those of 

 rectilinear vibrations ; and it vpas 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 Rev. Charles Graves read the first part of a paper on 

 Algebraic Triplets. 



The object which he proposes to hinaself 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 V^— 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 a; -|- v^ — 1 . y, in 

 which X and y are real quantities, and multiply it by a similar 

 couplet Xi-\- v — 1 . i/i, the product will likewise be a bino- 

 mial of the same kind, x.2-\- w ~ I .y-i; and between the 

 constituents of the three couplets there exists the relation 



(a;2 -H y^) (0^1^ + y,^) = xi + y^\ (a) 



But couplets may be more readily compared after undergoing 

 a simple transformation. Such an expression as a;+ \/ ~^.y 

 may be reduced to the form re^-'*^ by making r—Vx^-^y'^, 

 and B = tan-'[ -j. Hence it appears that if we agree to call 



r the modulus and 6 the amplitude of a couplet, the following 

 theorems will be true : 



