250 Mr. DE MORGAN, ON TRIPLE ALGEBRA. 



under the condition that ^^ means -y/- !> and if we then reduce the result to a line of the 

 same value of I, 6, w, in which also b = q, we have 



aa - pp' -2bb' - (pb' +p'b)\/2 + (ap' + ap)\/- 1 + (ab' + a'b) (1 + \/ - 1)^. 



Now for b ^2 write b, and let (1 + y/- 1)^ "^ \/~ be an independent unit symbol (it will be 

 found by our conventions to be a unit on the axis of sr), and for it write T; also for •%/— !> a 

 unit on the axis of y, write »j. Then it appears that the product of a + br] + ci^ (write c for p 

 and then interchange it with b in the preceding), and a + b't] + c'^ is 



aa - (6 + c) (6' + c') + (ab' + ba')n + (ac + ca')^, 



which is here produced, and can only be produced, from rf= —I, ^^= - 1, >;^= - 1. 



I shall give the interpretation of this synthetically, and with some minuteness, since the leading 

 features of it belong to most of the other imperfect quadratic systems which I have tried. 



Let every line drawn through the origin be considered as having for its plane that plane which 

 also passes through the axis of «; and let the line in which that plane cuts the plane of yz 

 be called the imaginary axis of that plane and of all lines in it (except the axis of ,t? itself). 

 Let a line (sr = - y) which bisects the second and fourth right angles in the plane of yz be called 

 the neutral axis, and one perpendicular to it, which therefore bisects the first and third right angles, 

 the primary axis. Let every imaginary axis have for its sign the sign of the parts of y and z 

 which lie on the same side of the neutral axis as itself: and let angles be measured positively 

 in every plane by revolution from the positive axis of x towards the positive imaginary axis. 



Let o + 6»j + c^ represent a line of the length Z = .^|a^+ (6 + c)^| in a plane whose imaginary 

 axis make with the positive axis of y the angle =tan~'(c : b) having for projections on the real 

 axis (the axis of x), and its own imaginary axis severally a and b+c; or making with the 

 axis of X an angle 6 whose sine is 6 + c : I and whose cosine is a : I. 



For addition, subtraction, multiplication and division, of two lines, make them both revolve 

 round the axis of x into, say the plane of wy, taking care to bring the positive part of each 

 imaginary axis into contact with the positive part of y. Then add, subtract, multiply and divide 

 as in common double Algebra, and find the plane into which the results are to be finally 

 transferred by the following rules. 



In addition, set off on the primary axis lines equal to the projections of the given lines on their 

 imaginary axes ; or transfer the imaginary projections by revolution to their proper sides of the 

 primary axis. From the extremities of the lines so drawn, draw lines perpendicular to the 

 primary axis, meeting the imaginary axes of ihe two lines, so as to cut off two hypothenuses. 

 On these hypothenuses describe a parallelogram ; its diagonal from the origin is in the imaginary 

 axis of the sum. And similarly for the subtraction, or the addition of the equal and opposite line. 



In multiplication, first lay down on the primary axis lines proportional to the tangents of 

 the angles which the factors make with the axis of x, and then proceed (exactly as in addition) 

 to determine the imaginary axis of the product from the diagonal of the hypothenuses. And 

 similarly for division. 



In every plane, as long as lines are taken in that plane only, there is one complete system 

 of double Algebra, admitting every rule of ordinary Algebra to its full extent. When lines 

 from another plane are introduced, we lose the equation A(BC) = {AB)C, unless A and B be 

 in one plane. 



The theory of powers and roots is absolutely identical with that of common double Algebra for 

 every line which is not on the axis of x, the plane of each line being the locus of all its powers. 

 And + 1 has only two square roots, as usual ; but — 1 has an infinite number of square roots, every 

 imao-inary axis of a unit in length being one of them. Also both + 1 and - 1 have an infinite 

 number of third, fourth, &c. roots, one set of three, four, he, in every plane. 



