Mr. J. Cockle o« the true Amplitude of a Tessarine. 291 



provided that in the expression for the true modulus M (which 

 may be seen at p. 4-35 of the preceding volumeof this Journal) 

 both the undetermined signs are taken as positive. Let M, 

 p^ q, rbe called the Elements of the tessarine t-, then, if M', 

 p', 5^', r' be the elements of/', and M", p'\ q", r" those of/"; 

 and, further, if /, /', and i" be connected by the relation 

 ti'=:t", the following equations will hold between the elements 

 of the product and of the factors ; viz. 



gr"cosj9"={l — (g, g')} cos^cosjo'— {I— (r, r')} sin^sinjs', 



7^' s\np"={l — {q,i'')} cosjf> sinjo'+{l— (^'r)} cosjo' sin^, 



(I — g") cosy = ((7, §'') cos^ cosp'— (r, r') sinp s'lnp'j 



{I— r") sin pf'={q, r') cosp ship' -{- (q', r) cos p' sin p, 



where {a, b) denotes a + b — 2ab. The above four equations will 

 readily be verified by a comparison with those which I gave 

 (at p. 437 of vol. xxxiii. of the Phil. Mag. S. 3) as connecting 

 the constituents of the product with those of the factors; add 

 the first and the third of them together, and we have 



cos 7>" = cos p cos jo' — sin p sin p' = cos [p +/»'), . (a.) 



whence, calling p the true amplitude of t, we infer that the 

 true amplitude of the product of two tessarines is the sum 

 of the true amplitudes of the factors. And we might have 

 inferred the same thing by adding the second and fourth of 

 the above equations, which would have given us 



sinp"= sin {p+p') (b.) 



The combination of (a.) and (b.) shows us, however, that, 

 rejecting circumferences, p"=p-{-p' is the only relation between 

 p, p', and p". It will be remembered, in the above investiga- 

 tion, that M"= MM'. 



I may add that these relations, both between the moduli and 

 the amplitudes of systems of tessarines, may be readily arrived 

 at by combining the reasoning of paragraphs 10 to 12 of the 

 Rev. Professor Charles Graves's paper on Triple Algebra at 

 pp. 1 19-126 of vol. xxxiv. of this Journal*, with that used by 



* In the same Number (Phil. Mag. for February 1849) I have given 

 (see pp. 132-135) a " Solution of Two Geometrical Problems," by means 

 of a Uniaxal Geometry. The rationale of the virtual solution is clear and 

 logical. Let it be required to find a point situate at a distance a from a 

 given point (the origin, for instance), and fulfilling certain (specified) con- 

 ditions. In Uniaxal Geometry the solution proceeds by supposing the 

 required point to be situate in the primary axis or axe — as to which latter 

 term see Phil. Mag. S. 3. vol. xxxiv, pp. 408, 409. And on this suppo- 

 sition we find for the distance of the required froni the given point an ex- 

 pression of the form 



A+B V^— 1 (or, more generally, A-j-B -v/^-f C/); 



