5, 6] VECTOR PRODUCT. VELOCITY. 9 



xm x + ym y + zm z = 

 s x m x + s y m y + s z m z = 0, 



that is, the vector product of the two vectors is perpendicular to 

 their plane. From the definition of moment, or by reference to Fig. 3, 

 its magnitude or tensor is equal to the product of their tensors times 

 the sine of the angle included between them. (It is to be noted that 

 the projections of the first factor in the vector product follow each 

 other in cyclic order in equations 12), those of the second factor in 

 reverse order.) It is at once evident that the moment of the resul- 

 tant of two vectors with the same initial point is the resultant of 

 their individual moments. Thus moments are to be considered in all 

 respects like vectors. It is evident that the moment of a vector, 

 Sx, Sy, s z , with initial point x, y, s, about a point |, ??, J, has the 

 projections: 



m x = (y- vj) s, - (e - Q s y 



13) m y = (z g) s x (x !) s z 

 m z = (x- fj) s y (y - 7?) s x . 



6. Velocity. As a second means of description of the motion of 

 a point we may give the geometrical locus of the positions that it 

 occupies at different instants. This is called the path of the point, 

 and if it is straight, the motion is said to be rectilinear. This alone 

 does not suffice to describe the motion, for the same path may be 

 described with different speeds. We must therefore give something 

 which shall determine what positions are reached at various instants. 

 If we call s the distance the point has traversed in its path, counting 

 from a fixed point, and give the value of s for every value of t 

 s = <p(f), this together with the equations of the path, which may be 



14) F,(x, y, e) - 0, F t (x,y,e)-0, 



completely specifies the motion, making as before three equations. 

 The velocity of the point is defined as the limit of the ratio of the 

 length of the path As described in an interval of time A to the 

 time A when both decrease without limit, that is, 



Velocities of the same numerical magnitude may however have 

 different directions, accordingly to completely specify a velocity we 

 must give not only its magnitude, but also its direction. It is there- 

 fore a vector quantity. Its direction is that of the tangent to the 

 path at the point in question, and its direction cosines are 



dx dy dz 

 ~d~8 y ~ds' ~ds' 



