Comparison shows that V x H = 4srC, where is the magnetic force at (x, y, is), and C is the electric flux through unit area per unit time, i.e., the current density; this is Maxwell's equa tion for electric current in a magnetic field.
32. Vectors in a Plane Quadrantal Ver Consider the vector r=ix jy, in the xy-plane, and perform the operation k x, then k x r=—'x iy; the operation repeated gives k (k r)=—Lr—jy=---r, hence (k s and (k x) = each of these is a sym bol for the operation of turtling a vector through a positive right angle in the xy-plane, and is called a quadrantal versor.
33. Versor for Angle V.— Writing r in the form r r (I cos j sing) r (1 cos + kxf sin 0)=---r (cos + sin 0 k s )1 (cos e 1/-1 sin 0)i, we see that r can be derived from the unit vector i by applying the operator r (cos + sin 0), which is the product of the tensor r and the versor cos 0 sin 0; this versor acting alone turns a vector through the angle 9 without stretching. The product of the versor for angle Sand the versor for angle se is the versor for angle tl le, thus (cos 9 -4 1/-1 sin e) (cos ei +v=r sin (P) =cos (9 + + V —1 sin (0J 0'). The complex algebraic number x y tarn be represented by the versi-tensor r (cos 0 1/-1 sin 9) where r =1/ tan 0= Yix• 34. Vectors and Versors for Simple Har monics.— Let the vector r, revolve with angu lar velocity 41 radians per -second; then Et•sa try where is the value of °when t = 0; and the a-component of ri is xi = ri cos (sit + 0). Let another vector r, revolve with the same angular velocity, starting at the position 0=02, then its x-component is xi =rs cos (wt 0i).
The vector sum (the diagonal of parallelo gram) is rii=r1+ r, whose x-componeat x ri cos (Gil + C) rs cos (ut + es); the diagonal r, remains constant to length and revolves with angular velocity (a, hence its projection is of the form x= cot (wt es), a simple harmonic function of t with period 21r/w; hence the sum of two simple harmonic functions with the same period and different phase angles is another simple hairs monk of the same period, and its character istic vector is the sum of their vectors. The phase angle es is found by adding the corn= ponent vectors at time t's-0.
35. Time Derivatives of Simple Harmonics. — If e = ei cos (wt + 0), then de/dt = sin (wt + 0) = cos (64 + 0 trr), whose characteristic vector has the angle 0 + fir, and is represented (32) by jwe, where j = 1(-1; similarly cfe/de has the characteristic vector that is, — w'e. This enables us to elimi nate the time variable from differential equa tions involving simple harmonics of the same period, as in the theory of alternating electric currents.
Bibliography.— For further details and ap plications consult the treatises of Gibbs and Wilson (New York 1907) ; Coffin (New York 1911) ; Burali-Forti and Marcolongo (Bologna 1909) • Bucherer (Leipzig 1905) ; also Heavis side, Theory) (London 1894); Abraham and Foppl, Theorie d. Elek' (Leipzig 1907).