Mr. DE morgan, ON TRIPLE ALGEBRA. 247 



only case of that system in which 1 = y/ {a- + kc), the equations (M) admitting no other solution 

 with that modulus. 



We now come to the question of geometrical interpretation, the most difficult part of the question 

 in one sense, the easiest in another. Every system of Algebra admits of an infinite number of 

 geometrical interpretations. Take the common one, and instead of supposing ,r + y ^- 1 to stand 

 for a line r = ^ (x^+ y^) inclined to the axis of x at an angle Q = tan "'(y : x), let it stand for 

 any line r, inclined at an angle Q^, where r, and d^ are unambiguous functions of r and 9. 

 Then the sign + in [r,, 0,] + [/i, Q'l] must be defined in such a way that the preceding symbol 

 may stand for the line determined by r = ^y \{x + x''f+ {y + y')-\ and tan 6 = {y + y) : {x + ,i') ; 

 and similarly with the other signs. There is no question about the superior convenience and 

 primary character of the usual interpretation : but others are not therefore absolutely excluded. 



Analogy would lead us to infer that a, 6, c should represent lines on the axes of ai, y, z ; 

 and even if we took them to represent areas on the planes of yz, zx, and xy, we should be 

 able to determine an area on the plane of yz (its form not being in question) by a line on the 

 axis of X. Again, the same analogy would lead us to take I for the absolute length of a + bri + ci": 

 but all that is necessary is that (, 6, and (p should be sufficient determinants of that length. For 

 instance, we may say, let a + brj + c^ represent a length r = -v/(a^+ b^ + (?) inclined to the axes at 

 angles having cosines X, /x, v, proportional to a, b, c : but then we give up the convenient property 

 of the modulus of multiplication, and must form (R, A, M, N) the product of (/•, }., m, v) and 

 (r'j X', fi, v) from the conditions 



.B cos A = rr' (/ii/'+ i'^' + XX'), 



B cosM = rr' (\iu'+ \'/jL - vv'), , 



B cos Is = rr' (Xv + X'l/ - fifx'), 



so that B must depend on the angles of the factors as well as on their lengths. The systems 

 I have given are the only ones in which the moduli represent the absolute magnitude of the 

 symbols. 



I am not able to present any striking geometrical interpretation. The symbols of the triple 

 trigonometry on which it must be founded are mixed functions of circular and hyperbolic sines and 

 cosines. If we take the equilateral hyperbola x'- y^= l, and let x and y be called the hyperbolic 

 sine and cosine of (p, the double of the sectorial area included between the axis o{ x, the radius vector, 

 and the curve (for analogy, the angle must be re|)laced by the double of the area of a circular 

 sector of radius unity), we have e* = COS (p + SIN cp, using capital letters for distinction. We 

 might very easily invent interpretations : but I see none which I think worth presenting. The 

 transformation 



cos 9 1 . „ 



± ^ sm = f cos (60" =F 9) 



will of course not be forgotten by any one who makes an attempt. This entrance of both species 

 of sines and cosines is, both in this and other cases, the consequence of the determination to have 

 what may be called a doubly logarithmic system, or one in which both angles, or magnitudes 

 corresponding to them, have their sums in the product. 



We may, if we like, consider the system as one in which there is a double modulus of mul- 

 tiplication ; let l.e* = m, and we have 



I - '\/(a'' + b' + c' + ab + ac — be), tit = a - h - c, 



a = § / cos 9 + ^m, a + i (/* + e) = / cos <?, 



6 = f / cos (fiO" -9) - J m, ^■s/3.(h- t) = I sine. 



p = §/cos(6W+ 9)-^m. 

 The product of [/, m, 9) and [(', m', ff] is now [I/', mm', 9 + 0'\. 



