374 APPENDIX. [297- 



297. Method of Dimensions. We can frequently determine the form 

 of a result by consideration of the dimensions of the quantities involved. 

 This will be made clear by the consideration of some examples. Thus, if 

 we assume that the period of oscillation of a pendulum can depend only on 

 its mass, its length, and the acceleration due to gravity, we can prove that 

 it is proportional to the square root of the length. Since the quantity to be 

 expressed is an interval of time its expression cannot involve any power of a 

 mass, and we have assumed that no mass but the mass of the body can 

 enter into the expression ; the period is therefore independent of the mass 

 of the body. Now g has dimension symbol [^J 1 ^]" 2 , and therefore lf\fg 

 has dimension symbol [^J 1 ^]"^ hence the only way in which the expression 

 of the period can contain the length I of the pendulum is by being propor- 

 tional to its square root. This argument would prove that the period is a 

 numerical multiple of \/(%). Again, to take another example, consider the 

 ellipticity of the Earth supposed to depend on the angular velocity of 

 rotation o>, the mean density p, and the constant of gravitation y. The 

 product yp has dimension symbol [T 7 ]" 2 , and thus o> 2 /#p is a number (angles 

 being measured in circular measure); the ellipticity being a number, must 

 be a function of <* 2 /yp. The method of dimensions supplies also a useful 

 means of verification. In any piece of mathematical reasoning where the 

 numbers represent quantities all the terms in each equation must be of the 

 same dimensions. 



298. Units. Throughout this book, except occasionally in examples, it 

 has been assumed that the unit of length is one centimetre, the unit of time 

 one mean solar second, and the unit of mass one gramme. This system of 

 units is known as the c. G. s. (centimetre, gramme, second) system. Some of 

 the derived units have received names which have met with general accept- 

 ance, such as the naines for the units of force and work. 



The C.G.S. unit of force is the dyne, it is the force which acting on a mass 

 of one gramme for one second generates in it a velocity of one centimetre 

 per second. The weight in London of a body whose mass is one gramme is 

 about 981-2 dynes. T'he gravitational attraction between two spheres each 

 of mass one gramme with their centres at a distance of one centimetre is 

 (6 '65) 10 ~ 8 dynes. This is the value of the constant of gravitation y in 

 C.G.s. units*. 



The c. G. s. unit of work is the erg, it is the work done by a force of one 

 dyne acting over a displacement of one centimetre. 



299. British Gravitation Units. The c. G. s. system, although generally 

 used for scientific work, is frequently not employed in practical applications 

 of science. Engineers in this country use a system of units adapted to their 

 purposes and associated with the commercial measures of quantities here 

 adopted. British tradesmen do not generally estimate lengths in centimetres 

 (or metres) nor do they sell things by the gramme (or kilogramme). The 



* C. V. Boys, Proc. R. S. Vol. LVI. 1894. 



