PHYSICAL CURVES. 97 



roadway of the bridge and its various loads are usually more 

 nearly distributed uniformly along the horizontal than along 

 the curve. As mentioned elsewhere, if the load were uni- 

 formly distributed along the horizontal and the cables of the 

 bridge possessed no weight, then the curve assumed would be 

 parabolic. But the cables possess weight uniformly distri- 

 buted along their length and the resulting curve assumed by 

 them is therefore neither a catenary nor a parabola. As 

 examples of catenaries may be mentioned the ordinary elec- 

 tric light wires, telegraph wires, etc., which hang in true cat- 

 enary curves, and of which the tension at any point can be 

 exactly calculated. 



The Logarithmic and Exponential Curves. — Logarithms were 

 originally invented solely for the purpose of shortening the ordi- 

 nary processes of multiplication, division, etc. The natural 

 numbers and their corresponding logarithms when plotted as 

 abscissas and ordinates constitute the logarithmic curve. 

 Although in origin logarithms are anything but physical, yet 

 the laws underlying logarithmic equations (and the closely 

 related exponential equations) have important physical rela- 

 tions. The logarithmic and exponential functions have as 

 their foundation an equation in which one of the variables 

 enters as the exponent of a constant base. 



An equation of this form expresses what is known some- 

 times as the compound interest law, as in a modified form it 

 can be used for the calculation of principal at compound 

 interest — a very materialistic relation between mathematics 

 and physics. 



If the exponent in the exponential expression is negative, 

 and the equation is properly modified, there results what is 

 known as Newton's law of cooling, expressing the tempera- 

 ture of a cooling body at any given time. Likewise, the law 

 of chemical reactions is expressed by an exponential equation 

 with negative exponent. Other examples of the same rela- 

 tion might be mentioned. 



These are all matters concerning quantities which increase 



