Mr. DE morgan, ON TRIPLE ALGEBRA. 251 



For a + bri + ci^, or [I, 6, lo], we may write 



I sin 9 I sin 9 . tan t» ^ 



/ cos H ri H . r, 



1 + tan (o 1 + tan w ^ 



and if a/„ — 1 denote the square root of — 1 which is at an angle w to the axis of y, we have 



, 1 tan u) y . V / 



^ 1 + tan te) 1 + tan w a v j^ 



Call these last -v/- 1 and -^j — 1 ; we have then 



r . ^ T w ^ • /I / ^ ,( n s'" ^ y sin tan w , \ 



[/, 0, col = /(COS0 + S\\\9 W^- 1) =nCOS0 + .a/- 1 + -i/, - 1 . 



'- -" ^ ^ V 1 + tan (o ^ 1 + tan (o *^ / 



The product of any Ufo positive square roots of - 1 is — I, and the product of a positive 

 and negative square root is + 1. 



The Algebra of the neutral plane, which passes through the neutral axis and the axis of cc is of 

 a very peculiar character. In the first place, neither side of the neutral axis is necessarily positive 

 or negative by our conventions, and the signs of this axis must be determined (like that of 

 tangTr) by the manner in which we come upon it. But this is not the chief peculiarity. If 

 we call the point whose co-ordinates are a, b, c, the subsidiary point of L or [/, 9, w], the 

 point and its subsidiary point are always in the same plane : but if the subsidiary point be on 

 the neutral plane (6 + c = 0), the angle is or tf, and L is on the axis of x. But if on the 

 other hand L be on the neutral plane, but not on the axis of a, then 6 and c are infinite (with 

 contrary signs): and in this case, whatever line A may be, L^J, A Js L, A x. L, A-r-L, L-i-A, 

 are all on the neuti'al plane. 



Hence 'a unit, situated on the positive side of the axis of «', is not a complete description of 

 any line : for under that description comes every case of 1 + m (»; — ^) in which m is finite. 

 The fundamental unit 1 or l + 0); + 0^is the line which requires that the preceding should be 

 augmented by ' having its subsidiary point at its extremity.' It is true that no alteration could, 

 in any case, be produced in I or 9, by substituting one case of 1 + m (rj — ^) for another ; but 

 the effect would be seen in the value of w. The rules of addition and multiplication, as above 

 given, fail when one of the lines is of the form a + t7iri — m^; we must replace them by others 

 drawn from the use of the projections themselves. 



I look upon the preceding system, as the one which has most general resemblance to the 

 common system, from which I derived it, before I considered the subject generally. 



It is demonstrably impossible that any system can give the convertibility of three factors, in 

 which aline of a unit in length is represented by cos + sin . P^, where P„P„. = — 1. Calling 

 this A, it will be found that A" A' A and A' A" A are not identical unless sin 9 . sin Q'. sin 9" . P„ 

 = sin . sin ff. sin 9". P„-, which, to be universal, requires P„. = P^.. 



^ 6. Second imjierfect system deduced from the redundant system. It is natural to examine 

 that particular mode of getting rid of redundancy, which consists in reducing the modulus of 

 multiplication to the form ■\/(a^ + J)' + b'^ + ef). This is obviously 



a(b - g) + p(b + f/) = 0, or b(p + a) + q{p - a) = 0. 



Now if we examine the corresponding function in the product, we find* 



A(B-Q)+{B+Q) 



= {(lib - ri) +;>(/;+ f/) \ \ a'- + b'- + p- + (/' \ + {a {!>' -(]') + p (/>' + r/') \ \ a- + h^ + p' +<?'}, 



• Mo«t caxily Keen thus : since 



i* tdentical with the product of the corresponding t'unctiuns of 

 o, i, &c. and a', 6', ice, the parts afleclcd with v'- arc identical; 



whence follows tlic equation in the text, and also 



. +2{a(4-(7) + p(4 + v)|la'(4'-7')l + ip'{4' + 7)l. 

 K K 2 



