252 Mr. DE MORGAN, ON TRIPLE ALGEBRA. 



So that if this condition be true of the factors, it is true of the product. Now if as before, 

 a + P\/- 1 = re"'*'"', b + p ■s/ - 1 = Se^*^"', we have, for the expression of the condition, 

 tan (/3 - a) = 1. This gives either 



/3 + \ir = a + \ir, I cos 9 = r cos a — s sin a, I smQ = r sin a + s cos a, 



or /3+^7r = a + ^7r, I cos Q = r cos a + s sin a, ^ sin 9 = r sin a - s cos a. 



The first will be the most convenient. 



But though this condition may be satisfied for the product, when it is so for the factors, the same 

 is not true of tlie components and the sum, unless a : a :: p : p' . This system then would be 

 perfect for multiplication, division, and all its consequences, as the former one is for addition and 

 subtraction. 



If we endeavour to find the system in which the sum of two lines is the diagonal of the 

 parallelogram formed on them as they stand, at angles a and /3 + ^tt to the axis of x in the two 

 planes; we find the condition to be p{h + q) = 0. Now b - - q satisfies this for additions, and 

 jo = and b = - q for both additions and multiplications : but an examination of this last case 

 will shew that it gives nothing more than the common double Algebra ; no line lying out of 

 the plane of wy. 



If there can be a perfect non-redundant system formed out of the redundant system, there must 

 be some function f(a, b, p, q) such that f{A, B, P, Q) and /(a + a, b + b', p + p', q + q) both 

 vanish when f(a, b, p, q) and /(«', b', p, q) both vanish. The second condition cannot be satis- 

 fied unless / (a, b, p, q) be of the first degree with respect to the letters specified, in which case 

 the first condition cannot be satisfied. 



& 7- Imperfect system, independent of all that precede. Let the laws of combination of 

 the symbols, ^, >;, ^, in the expression a^ + bt] + c^, be 



The product of of + bij + c^ and a'^ + b'ri + c'^ is 



\aa' - (6 +c) (i' + c')}f + {ac +ca'| tj + \ab' + ba']^. 



In this system, the properties of the neutral and primary axes, the conventions of sign connected 

 with them, the modulus of multiplication, the rule of addition and subtraction, and the meaning 

 of the angles 6 and w, are precisely as in the system described in j 5. But the product of 

 two lines in this system differs from that in the preceding one as follows ; the angle made by 

 its imaginary axis with the axis of y is the complement of that made in 5 5. Or, signifying 

 by (p the angle made by [/, 0, <p] or a^ + btj + c^ with the primary awis, then if [;, 9, A] and 

 [I', 9', ^'] have the product \_L, 9, <1>] in § 5, their product is [L, 9, - 0] in the present system. 

 Let two imaginary axes be called opposite which are equally inclined to the primary axis on 

 opposite sides of it, and let the planes passing through them and the axis of x be called opposite 

 planes. Then A"" is in the plane opposite to that of A""; A*, A'"', A", &c. are in the plane of 

 A; A'', A", A^^, &c. are in the opposite plane. Generally speaking .4^" + ' is in a new plane for 

 every new value of to. But the character of the square roots of — f resembles that in & 5, and we 

 have 



[l, 9, ^] = /{cos0.f + sin0v'*-^l 



lcos9 -t + I sin9. ■^ . » + Ism 9 ^ C- 



