Division is interpreted by introducing the reciprocal, defined by the equation qr =1, in which q and r are said to be reciprocal to one another, and, as in ordinary algebra, we write r = and g•=.t---'. This product obviously obeys the commutative law and also it has no vector part, so that x' = nx, I ny, s'=nz, En= a number] and "MX == — te X , tti "= -- ntv.
Hence the reciprocal of q must have the form 1 q—'= (le — xi— where m is a number. By forming the product observing the laws i =1 = k' =_1, 1.1=—ji= k, etc., it is easily shown that m= le + x' + y' + a'.
This number is called the norm of q.
It is now evident that if r -= w' + x'i + y'j + z'k and q= w + xi* yj + zk, the quotient is obtained by writing out the product 1 (w' ± x'i + y'j + Jek . in --w—xi—yj—zk), and that this quotient is a quaternion.
Hence, barring critical cases, such as division by zero, indeterminate forms, etc., the applica tion of 'the four fundamental processes of algebra to quaternion symbols leads always to determinate quaternion results. The quater nionic algebra is, therefore, a closed system and satisfies the definitions of a group with respect to addition, multiplication, subtraction or divi sion. See GROUPS, THEORY OF.
The positive square root of the norm of q (= + Jw' + x' + + ts') is called its tensor and the quotient of q by tensor of q is called its versor, so that a quaternion is always the product of its tensor and versor. The sym bolic form of this statement is g=TTLIg, where T and U stand for tensor and versor respectively.
In the strictly symbolic notation the recip rocal of q has the form Sq — , and since it follows that Sig— Vlq=7‘g.
The quaternions Sq + Vg and Sq — Vq are said to be conjugates of one another.
A vector has no scalar part, but it has a tensor factor and a versor (vector of unit length) ; and a scalar has no versor part, but it has a tensor, its positive numerical value, and a versor whose value is always ± 1. it can readily be shown that the square of the versor of a xi vector is-1. Let the versor be _ , yj sk v x 2 + + Its square will be ya ± ZXZX +k xy—xy 1 0 + Y2 + I - -Fsq The symbolic statements are Vg=TVg.UVq,
Sq=± TSq.
Therefore, also, V'q--TVq, TSg.
The equation S'q — V'q = T'q may now be written + and hence, 8 being an appropriate angle, it may be assumed that Sq=Tq.cos 0, Vq.....Tq.UVq.sin 0.
By the introduction of these expressions for scalar and vector the quaternion is presented in a new binomial form q= Tg(cos + UVq.sin 0), one of great importance and utility.
The formula last written is a particular case of a more general geometric law. A recent interpretation makes quaternion symbols repre sent straight-line segments (their directions include) in four-dimensional space.* Inter preted in this way quaternions are called directors. It is shown that if two quatemion directors be perpendicular to one another their quotient is a vector. Let 0 be the angle between p and q, let s be the director perpen dicular dropped from the terminal extremity of q to p, and r the director from the origin (intersection of p, q) to the foot of this perpen dicular. Then by geometric addition and q/P = r/P s/p; and s/p, being the quotation of a pair of mutu ally perpendicular directors, is a vector. Hence, since if p be the ratio Us/Up, it is a unit vector, Tq gip= — Tp (cos p sin 0).
When p =1 this equation reduces to the one previously written for the value of q. In this particular case 8 becomes the angle formed by the quaternion director q with the axis of teal quantities, or scalar axis. When p and q are both vectors the formula expresses the fact, on which Hamilton early places emphasis, that the ratio of two vectors is in general a quaternion, but that if the vectors meet at a right angle the ratio is another vector, if they be parallel it is a scalar. But the formula also shows that these statements are true of quater nions in general.
The two parts of the equation last written present the quaternion under its three most fundamental aspects: 1. It is the ratio of two directed, non posited,* straight-line segments.
2. It is the sum of a number and a directed, non-posited vector.
3. It is the product of a tensor (tensor of q/p) and of a versor, the latter being expressed in terms of its angle 8, and its vector axis p.