Mr. J. Cockle on a new Imaginary in Algebra. 47 



6. Of the Modular Relations of Tessarines. 



Let 9 be the amplitude, f the colatitude, vj/ the longitude, 

 and IX, the modulus of a tessarine (t). To these quantities (which 

 are identical with the corresponding ones in the quaternion 

 theory, and which may, without confusion, be adopted from 

 that theory) must be added another, which I propose to call 

 the submodulus, and to denote by v. The submoduhis is de- 

 fined by the equation 



and the submodulus of the product (^") of two tessarine fac- 

 tors {t and t') is determined by the relation 



and the modulus of that product by 



We have, further, 



WTO)" + xa^^ + yy" + zz^' = ttj'jtt^ + 2y'v\ 

 and 



The equation for the submodulus may also be expressed as 

 follows: — 



v2=j«-2 sin 6 sin <p (cos d cos \l/ + sin 3 sin if/ cos <p). 

 The construction of this last equation, and of the preceding 

 ones, 1 hope to discuss on another occasion. And there is a 

 surface — that defined by the equation 



jM,?/ cos ^-\-x% = 

 (where % is supposed constant), — which appears to merit at- 

 tention in connexion with the theory of tessarines. 



Second Postscript, I'Mh December 1848. I have the satisfac- 

 tion of adding, that the hope above expressed has not been 

 disappointed, and that I have solved the problem of the four 

 equidistant points in the manner I proposed. I have deter- 

 mined the position of the fourth point by an expression of the 

 form 



A + m+iC; 



where C is to be measured in a direction perpendicular to the 

 plane in which the first, second, and third points are situate. 

 The four points are, of course, at the angles of a regular 

 tetrahedron. I hope to obtain for the solution a place in this 

 Journal. 



