QUATERNIONS. Fundamental Prin ciples. A quaternion, or of four,* is a quadrinomial of the form w + xi -I- yj sk, in which w, x, y, x are numbers, and 1, 1, j, k are four independent units, any three of which may be interpreted geometrically as a set of mutually perpendicular straight lines of unit length in three-dimensional space. The binary products xi, yj, sk are three mutually per pendicular line-segments of lengths x, y, respectively; they are called vectors, while pure numbers, positive or negative, are called scalars.

The calculus of quaternions is an algebra in which the fundamental operations of addi tion, subtraction, multiplication and division, and the consequent operations of involution, evolution, etc., are employed, and whose ele ments (operators and operands) are quater nions. It is a multiple algebra, because a quaternionic symbol, as q, contains implicitly several independent quantities, and it is essen tially geometric, because its operations, addi tion, multiplication, etc., may be interpreted as geometric transformations. For example, in illustration of the last statement, if q, r be two quaternions and be taken to represent two line-segments meeting at a point, their sum is the diagonal of the parallelogram of which q and r are two adjacent sides. This is known as geometric addition; its law is that of the parallelogram of forces. Multiplication by a quaternion is equivalent to an orthogonal transformation in four variables plus an ex pansion or a contraction. The geometrical construction of a quaternion product, however, needs the hypothesis of a four-dimensional space for its proper presentation.* Vectors in particular are so defined as to obey the law of geometric addition. If several vectors represent the successive parts of a broken line in space, their sum is the line segment joining the free ends of the broken line. In this process of addition the line segments representing vectors may be trans ferred to any position in space, provided they remain parallel to their original directions, but the direction-sense must in every case be preserved.

In obedience to this interpretation a qua ternion may always be written as the sum of a scalar and a three-dimensional vector; thus, q=..Sq Vq,

wherein S and V are to be read ((scalar of* and ((vector of* respectively; for by the law of geometric addition xi + yj zk is itself a vector, while w, being a pure number, is by definition a scalar.

In the quaternion algebra the laws of asso ciation and distribution in the four funda mental operations, and the law of commuta tion in addition and subtraction, are assumed, or they may he derived from the geometrical definitions of the processes, but the commu tative law in multiplication and division is rejected except for such quaternions as have vector parts that are numerical multiples of one another.

The laws of combination by multiplication are determined in the following manner: The See " Bulletin of the American Mathematical eccietY " (Vol. 11).

mutually perpendicular vectors xi, yj, zk are numerical multiples of the three independent units, i, j, k, and the combinatory laws for these units are derived from the assumptions ? f ks= 1, ijk = 1, from which are obtained, through multiplica tion by i, j, k in succession, the remaining binary products jk=kj=i, ki-=ik= j, ij= ji=k. From these laws it follows that the product of two quaternions is itself a quaternion. Thus q and r being given quaternions, qw + xi+ yj + zk, r =. w' ฑ x'i + yi+ z'k, their product has the form qr = W + Xi + Yj +Zk, wherein W = wtv' --- xx yy' zz', X =tux' + w's + yz' 1z, Y.= wy' + w'y+ zx' 2'x, Zฎ we' + viz + xy' x'11.

If the order of the factors in this product be changed from qr to rq, the scalar part ww' xx' yy' zz' is unaltered, but the terms yz' ytt, zx' s's, xy' x'y change their algebraic signs and, therefore, the vec tors of qr and of rq are different. This result shows that in general quaternion multiplication does not obey the commutative law. In order that the above terms may not change sign (by a reversal of the order of the factors q, r) they must be separately zero, that is, zsz xxz ;" P y" in other words, x', y', s' must be the same numerical multiples of x, y, z respectively. This makes x'i +1; + z'k a numerical multiple of xi + yj + zk, and it is the condition under which it is permissible to write qr =rq.