Dimensions of Physical Quantities. 107 



is a given concrete acceleration. The only practically im- 

 portant cases are those of force, work, power, and the 

 mechanical equivalent of heat, and if we multiply the number 

 expressing any one of these quantities in a gravitation system 

 by (the abstract number) y, and remember the condition that 

 7 [L T~2] is constant^ transformations can be effected as 

 usual. 



Thus, if J, J', and J" be the numerical values of the 

 mechanical equivalent expressed in an absolute, and two 

 gravitational systems, we have 



J [L' T-2 6'-!] = J' y [L'2 T'-2 ^'-1] = J" 7" [L"2 T"-2 $<'-']. 



Thus to find the numerical value of J in the C.G.S. system 

 when it is given in the English gravitational system, 



J = 773-2 X 32-2 (JS^Y / degree Cent^>^..^g ^ ^^,^ 

 V centim.y v degree rahr. / 



To find J in the metre-kilogram-second gravitational sys- 

 tem, when it is given in the British gravitational system, we 

 reduce the last two of the above expressions by the relation 



y [L^ T'-2] =Y'\_L" T''-2] to the form 



J'[L'6'^-i]=J''[L''r-i]; 

 so that 



J=773-2 



/ foot \ / degree Cent.\ ^ ^^^^ 

 \ metre/ \degreeFahr./ 



To reduce power from an absolute to a gravitational system, 

 or vice versa, we have 



n, [Ml W Tr^] =n, [72 Mg L/ T^'] • 



Thus to find the number of ergs per second which corre- 

 spond to one H.P., 



n = 



55500 x32-2/P°^^V^Y=7-46xl09. 

 bU \ gram / \ cm. / 



It is noticeable that although the statement that J is inde- 

 pendent of the unit of mass is in general true, it holds good 

 only when the unit of heat and the unit of force are defined 

 with respect to the same unH of mass. In a gravitational 

 system this is not really the case, e.g. in the English system 

 the unit of heat is defined with reference to the j)ound, while 

 the employment of the gravitational measure of force involves 

 a unit of mass = 32*2 lbs. Hence, even when the nominal 

 unit of mass has disappeared, the number which expresses 

 the ratio between the real and nominal units on the gravita- 

 tional system remains in the expression for a quantity such 



