STEP-FUNCTIONS 7/2 



And, to complete the set, we need (D) the null-function, which 

 shows no change over the whole period of observation. The four 

 types obviously include all the possibilities, except for mixed 

 forms. The variables of Fig. 2/10/1 will be found to be part-, 

 full-, step-, and null-, functions respectively. 



In all cases the type-property is assumed to hold only over 

 the period of observation : what might happen at other times 

 is irrelevant. 



Sometimes physical entities cannot readily be allotted their 

 type. Thus, a steady musical note may be considered either as 

 unvarying in intensity, and therefore a null-function, or as 

 represented by particles of air which move continuously, and 

 therefore a full-function. In all such cases the confusion is at 

 once removed if one ceases to think of the real physical object 

 with its manifold properties, and selects that variable in which 

 one happens to be interested. 



7/2. Step-functions occur abundantly in nature, though the 

 very simplicity of their properties tends to keep them incon- 

 spicuous. ' Things in motion sooner catch the eye than what 

 not stirs '. The following examples approximate to the step- 

 function, and show its ubiquity : 



(1) The electric switch has an electrical resistance which 



remains constant except when it changes by a sudden 

 jump. 



(2) The electrical resistance of a fuse similarly stays at a low 



value for a time and then suddenly changes to a very 

 high value. 



(3) The viscosity of water, measured as the temperature 



passes 0° C, changes similarly. 



(4) If a piece of rubber is stretched, the pull it exerts is approxi- 



mately proportional to its length. The constant of 

 proportionality has a definite constant value unless the 

 elastic is stretched so far that it breaks. When this 

 happens the constant of proportionality suddenly 

 becomes zero, i.e. it changes as a step-function. 



(5) If a trajectory is drawn through the air, a few feet above the 



ground and parallel to it, the resistance it encounters as it 

 meets various objects varies in step-function form. 

 81 



