92 THEORY OF NEWTONIAN FORCES. [FT. I. CH. I. 



no idea of direction are called scalar quantities, for they may be 

 conceived as arranged on a scale according to their magnitude. 

 Such are time, temperature, size, density. 



Quantities whose specification involves the idea of direction as 

 well as of magnitude are called vector quantities. They may be 

 represented by geometrical directed lines, and all that has been 

 said of vector quantities and their addition, etc. applies to them. 



48. Degrees of Freedom. A set of magnitudes or para- 

 meters which completely specify a quantity are called its co- 

 ordinates. The number of coordinates required is called the 

 number of degrees of freedom of the quantity. For instance, 

 a point in a plane may be defined by two rectangular, or two polar 

 coordinates, and has two degrees of freedom. We may also say 

 that there is a double infinity or oo 2 of points in a plane. A point 

 in space requires three coordinates of any sort, and has three 

 degrees of freedom. Every independent relation that the coordi- 

 nates of a quantity are made to satisfy diminishes the number of 

 its degrees of freedom by one. For instance, a relation between 

 the rectangular coordinates of a point restricts it to lie on a 

 certain surface, it then has two degrees of freedom instead of 

 three, and requires but two coordinates to specify it. For example, 

 a point satisfies the condition # 2 + y 2 - + z* = a 2 . It lies on the 

 sphere of radius a, and may be fully specified by giving its lati- 

 tude and longitude. 



For the coordinates of a vector R we may take its projections 

 on the three coordinate axes. If we choose its length, or modulus, 

 and its three direction cosines, 



a = cos (Rx), /3 = cos (Ry), 7 = cos (Rz), 



one of the four coordinates R, a, & 7 is redundant, for the latter 

 three satisfy the identical relation 



This furnishes us an example of the general case where we 

 give n coordinates of a quantity satisfying k independent identical 

 relations, or equations of condition. The quantity then has only 

 n k degrees of freedom, and we may find n k independent 

 coordinates which completely specify it. 



