Mr. DE SIORGAN, ON TRIPLE ALGEBRA. 249 



Let ^ signify revolution through 45" in the plane of xz, so that if a+p^l = re°^~', 

 b + g \/— I = se^^'^, we have (b + q \/- 1)^ signifying a line s at an angle /3 + jtt in the 

 plane of xz. Moreover we have 



/cos0 = rcos a + scos (/3 + ^tt), ^sin = r sin a + s sin (/3 + jtt), 



so that the way to find I and 9 geometrically is as follows. In any plane, say that of {vy, 

 set off }• and s at angles a and /3 + ^tt: the diagonal of the parallelogram on these lines represents 

 the length I inclined at the angle 6 to the positive axis of x. In various systems I find that 

 when I sin 6 has the form M ± N, one of the simplest interpretations consists in making N = M tan w, 

 where w is the angle which the plane of the line and the axis of ,v makes with the positive side of 

 the plane of xy. In the present instance, this will give 



b + q b — q I sin 6 / sin tan to b + q 



tan (0 = ;- , Icosd = a + — j— , = p, = — — . 



P\/2 y/2 l+tanoj 1 + tan to ^2 



Here p, b, q can be found so as to give [I, 6, oi] for any given value of a. The system is now 

 complete, all the rules of Algebra are true of it, and it only remains to give the results their 

 easiest geometrical form. The most natural mode of proceeding is to examine the mode of escaping 

 redundancy, which consists in assigning one relation between a, b, p, and q. The case o{ b = q will 

 appear exceedingly remarkable, when viewed in connexion with the imperfect system which I shall 

 describe in the next section. 



According to our conventions, a+p-^- 1+6(1 + \/-l)t^ represents a line of the length 

 / = ^ ^a^+ (j9 + 6/y/2)-} inclined at an angle having a : / and (p + by/z) : I for its cosine and 

 sine, with a projection on the plane of yss which makes the angle tan"'|6.Y/2 : p\ with the positive 

 axis of y. But the relation B = Q does not obtain in the product ; and if we bring it about by a 

 proper use of our redundant letters, so as to represent the product [Z,, 9, Q] under the form 

 V + W^-l + X (\ + '^- \)'(^, we shall find that we have sacrificed the equation A(BC) 

 = (AB)C, which is no longer a formula of the Algebra. Owing to the redundant letter, two lines 

 may be identical in position, but must not therefore be considered as identical. Now the introduc- 

 tion of an equation of condition between a, 6, p, q, and the alteration of the product in such a 

 manner as to satisfy this same condition, is, in point of fact, the substitution for the product of a 

 line equivalent in position only. 



I shall resume this subject in the next section: but in the first place, observe that the modulus 

 admits of resolution into the square root of the sum of two other squares, namely 



Take another angle k such that 



p + a p - a 



/ cos K = b + — y— , Ism K = q + — ; — . 



-v/2 ^/2 



This angle k is not a new directing angle, being in fact O-^ir; and ^'^ is - ^- 1. 



The modes of interpretation will be better seen, so far as they are easily practicable, in the next 

 section. 



§ .5. Imperfect form, derived from, the preceding. The first system of triple Algebra which I 

 obtained was that in which P = a + b>i + cT, where iC, rf, and tj^ severally re])rcsent - I. I did not 

 at first see that though tiiis will give PJ^=P'P, it will not give P' (P P) = (P" 1^) P, except in 

 particular cases ; though it should have been obvious that »}^^, for instance, is not the same thing 

 OS i'l^)l- Now this is precisely the case of the redundant system already noticed, in which b = q. 

 If we multiply together a + p ^- 1 -i- h{l + y/ - 1)^ and a + p'^/ - \ + b\l + y/ - \)'^, 



Vol.. VIII. Paut III. Kk 



