If q be any other quaternion, it may be proved in a similar way that But < (91 +92) + 293 > < (91 +92 +93). therefore, a fortiori, 291+292>< (91 + 2); 291 + 292 + / 93 > < (91 + 2 + 93). As this process may be carried on to any extent with similar results, we may infer that In order, therefore, that OD = OB, we must have Evidently, then, 0 COS = ; or, = 120°. U (q + q') = Uq + Uq', only when Tq = Tq', and 0 = BOC = (1) (2) 2π i.e., in a special 3' The result of 53° to 56° is, that the symbols S, V, and K are, while T, 4, and U are not, distributive in the addition and subtraction of quaternions. CHAPTER VIII MULTIPLICATION AND DIVISION OF TWO QUATERNIONS SECTION 1 Diplanar Quaternions 57°. Before proceeding further it is necessary to explain the meaning of certain forms of expression that will be met with in the present and succeeding chapters. Sq192 means the scalar of the product q192. Similarly, Kq192 means the conjugate of the product 192. It does not mean the conjugate of q multiplied into 92, which will be written Kq1. 92, or, Kq1(2), or (Kq1)92. And so on for the other symbols. Points and brackets should never be omitted if their omission is likely to lead to any misapprehension. The product of any two quaternions is a quaternion. The quotient of any two quaternions is a quaternion. (1) = T T 58°. The tensor of the quotient (or product) of any twoquaternions is equal to the quotient (or product) of the tensors of the two quaternions. G For, let the two quaternions, reduced to a common denominator, be Then, ΤΥ = Ty Ty Ta = = TY тв Ται. B ᎢᏰ Τα Τβ Τα Τα Tq 2 Again, if the two quaternions be reduced to the forms, (1) α Τδ Τα ΤΟ Equation (2) embodies Euler's theorem, that the sum of four squares may be resolved into two factors, each of which is the sum of four squares. For the tensors of the quaternions 91, 92, may, 39° (5), be thrown into the form, respectively; and the product, 192, is some quaternion, say, 2 (W2 + X2 + Y2 + Z3) = (w12 + x12 + y12 + z12) (w ̧2 + x22 + Y122 + ≈12 2). 59°. The versor of the product (or quotient) of any two quaternions is equal to the product (or quotient) of the versors of the two quaternions. = 60°. Let q ( Sq + Vq) and r(= Sr+ Vr) be any two quaternions. = Then, qr SqSr+ SrVq + SqVr + VqVr, rq = SrSq+SrVq + SqVr + VrVq ; or, resolving the quaternions VqVr and VrVq, qr = (SqSr+ S. VqVr) + (SrVq + SqVr + V . VqVr)..(1) rq = (SrSq + S. VrVq) + (SrVq + SqVr + V . VrVq).. (2) Since the right hand member of both of these two equations is the sum of a scalar and a vector, we have a fresh proof that the product of any two quaternions is a quaternion. (a). From (1) and (2) we have i.e. unless the planes of the two quaternions are at right. angles. (b). From (1) and (2) we also have Adding and subtracting the equations of (4), = Vqr+ Vrq 2 (SrVq + SqVr) = (SqSr+S. VqVr) — (SrVq + SqVr + V . VqVr) = SqSr - SrVq - SqVr+ (S. VrVq + V. VrVq) = SqSr (Sr = KrKq = (1) gr+ Kqr2 Sqr = 2 (SqSr + S. VqVr)... (2) (c). S. qKr=S. (Sq + Vq) (Sr-Vr)=SqSr-S. VqVr; S(Kq. r) S. (Sq - Vq) (Sr+ Vr)=SqSr-S. VqVr; therefore, S . qKr = S(Kq. r). . (3) 63°. (a). Since qr ‡ rq, 60° (6), it follows that the multiplication of diplanar quaternions does not obey the commutative law. (b). The distributive law applies to the multiplication of quaternions. For, if we take four quaternions, p, q, r, s, in the quadrinomial form, 35°, it will be found by actual multiplication that (p+q) (r+8)= pr+qr+ps+qs = pr+ps+qr+qs = &c. The distributive law, therefore, applies to four quaternions. (c). If we actually multiply the product pq into r, the result will be found to be equal to the result of multiplying p into the product qr. The associative law, therefore, applies to three quaternions. It may be similarly shown that the multiplication of any number of quaternions is distributive and associative. |