XX INTRODUCTION. 



have been called by Prof. James Thomson "change ratios," which seems an 

 appropriate term. The term " conversion factor " is perhaps more generally 

 known, and has been used throughout this book. 



Conversion Factor. — In order to determine the symbolic expression for the 

 conversion factor for any physical quantity, it is sufficient to determine the degree 

 to which the quantities length, mass, and time are involved in the quantity. Thus, 

 a velocity is expressed by the ratio of the number representing a length to that 

 representing an interval of time, or L/T, an acceleration by a velocity-number 

 divided by an interval of time-number, or L/T'^, and so on, and the correspond- 

 ing ratios of units must therefore enter to precisely the same degree. The fac- 

 tors would thus be for the above cases, ///and Ijt"^. Equations of the form above 

 given for velocity and acceleration which show the dimensions of the quantity in 

 terms of the fundamental units are called " dimensional equations." Thus 



E = ML'^T-^ 



is the dimensional equation for energy, and ML'^T"^ is the dimensional formula 

 for energy. 



In general, if we have an equation for a physical quantity 



Q = CL^M^T^, 



where C is a constant and LMT represents length, mass, and time in terms of one 

 set of units, and we wish to transform to another set of units in terms of which 



LMT 



the length, mass, and time are L,M,T,, we have to find the value of ~,~,p^, which 

 ° ' ' ' LMT 



in accordance with the convention adopted above will be / m /, or the ratios of 



the magnitudes of the old to those of the new units. 



Thus Ly = L/, My = Mw, T;=:T/, and if Qy be the new quantity-number 



Qy = CLy^My'-Ty^ 



= CL"/»M*»z''T7' = Q^m% 



or the conversion factor is l"m^^, a quantity of precisely the same form as the 

 dimension formula L^M^T". 



We now proceed to form the dimensional and conversion factor formulae for 

 the more commonly occurring derived units. 



1. Area. — The unit of area is the square the side of which is measured by 

 the unit of length. The area of a surface is therefore expressed as 



S = CL'', 



where C is a constant depending on the shape of the boundary of the surface 

 and L a linear dimension. For example, if the surface be square and L be the 

 length of a side C is unity. If the boundary be a circle and L be a diameter 

 C = 7r/4, and so on. The dimensional formula is thus L\ and the conversion 

 factor l'\ 



2. Volume. — The unit of volume is the volume of a cube the edge of which 

 is measured by the unit of length. The volume of a body is therefore expressed as 



