qVatebnions. 598 QTIATEBNIONS. branch of pim- maUwnmiWs, it fiiuls numerous application^ in i.liysies. Tl,e «-•** ^""'^'^Pt J'f "" liar to the quaternion theory is that oi vecto,. A line-segment AB has not only length but ° - also direction, and two line-segments AB, A'B' are considered equal when they have the same absolute length and the same direction, e.g. in the parallelo- gram ABB'A', AB, A'B' are called recforst( Latin. vectores, carriers ) , be- cause they are consid- ered as 'carrying' the points A, A' to the ^ points B, B', respective- ly. It is therefore evident that a vector may be transported parallel to its original position with- out alteration in character, and henoe that it may be con- sidered as a symbol of translation. The sum of two vectors, AB and BC, is con- sidered to be that * vector which carries A to C, viz. AC. This does not mean that the absolute value of AB plus that of BC equals that of AC, but. that (direction being also considered) a force AB plus another ^ BC, or AB + AC in the figure, equals the force AD. It is there- fore evident that, with this definition of addition, the sum of the sides of a tri- angle or of any other closed polygon, con- sidered as vectors, is zero. Therefore, if we have three given rectangular vectors, OX, OY, OZ. and OP, any other vector, OP, can be resolved into tliree vec- tors respectively, parallel to (hence equal to parts of) OX. OY. OZ. These are RQ, QP. OR : for OR + RQ = OQ, and OQ + QP = OP. as above ex- ]ilained. If, now. we lay nir units on OX, OY, OZ, an<l designate them re- spectively by i,, i™, I3, or. as is more common in English works, by i, j, k, and designate OP by p. we shall have p = 'xi + i/j + ~k, the absolute leng'th of p being V^ + J/' + s'- These geometric ideas are elementarv, and had already been used by Mfibius (q.v.) irf his barycentric calculus before" Hamilton invented quaternions. A few illustrations of their use will be of value. Let OABC be a parallelogram, the diagonals inter- secting at X: then OX -f- XA = OA = CB = CX +XB, .■.OX-XB = CX-XA; but vectors cannot be equal unless parallel, and OXB intersects CXA; hence in the last equation it is necessary that OX — XB = = CX — XA, and hence that OX = XB and CX = XA. It is thus proved that the diagonals of a parallelogram bisect each other. Suppose a, (3 are two adjacent sides of a parallelo- gram, and X a line joining tlie raid-points of two opposite sides, then it is required to prove that x is parallel and equal to a. Drawing p it is seen that p =- + X — a +L, whence x = a, which (con- sidering x and a as vectors) shows them to be equal and parallel. Consider, now, what is meant by the ratio of two vectors OA and OB, or rather on how many distinct numbers this ratio depends. To change OA into OB requires in general, ( 1 ) a variation in length, or the ai)plication of a stretching factor; ('2) a variation in direction, which re- quires three angles. Hence the required ratio depends upon four distinct numbers, whence the name quaternion. The term may be defined as a number that alters a directed line-segment in length and direction. The stretching factor is called the tensor and is indicated by prefixing T to a quaternion. The turning factor is called the versor and is indicated by prefixing U. A scalar is a quaternion whose product lies on the ' same line as the multiplicand, and hence is merely a positive or negative numl)er. A vector is a quaternion that tunis 'J0°, or },it. The sym- bol for tensor of a quaternion is Tg; for versor of a quaternion, j Ug ; for scalar, S^ ; for vector, V(jf. The conju(jii1e of a quaternion q, writ- ten K(/, has the same tensor, plane, and angle, but the angle is reversed. Suppose I, J, K to be unit lengths on rectangular coor- dinates as in the figure. These are ( counterclockwise ) so situated that a positive ,^v,..,..^...v... ^^, rotation, through 90°, of J aliout I brings J to K; a similar rotation about K brings I to J; a similar rotation about J brings K to I. Call the operator that turns K into J, i; i.e. t = -f,or iJ = K. Similarly, I J let i = f^, or ;K = I ; and fc = r or kl = J. •^ J . K _ . i It therefore follows that— g = ^'~ I — >'~ i = 7,-. Hence— J = iK = i (iJ) = iJ% or —1 = i'. Similarly —1 = f , and — 1 = fc'. Also, since iK = ; ( '/I ) = — ijl, and iK = — J — — fcl, it follows thai. — ijl = — k, or that ij = k. Simi- larly jk = i ki = j. A similar line of reasoning
