XXXIV INTRODUCTION 



of light scattered from small particles, in so far as it depends upon the wave 

 length, reasons as follows : 1 



"The object is to compare the intensities of the incident and scattered ray; for these will 

 clearly be proportional. The number (/) expressing the ratio of the two amplitudes is a function 

 of the following quantities: — T, the volume of the disturbing particle; r, the distance of the 

 point under consideration from it; X, the wave length; b, the velocity of propagation of light; 

 D and D', the original and altered densities: of which the first three depend only on space, the 

 fourth on space and time, while the fifth and sixth introduce the consideration of mass. Other 

 elements of the problem there are none, except mere numbers and angles, which do not depend 

 upon the fundamental measurements of space, time, and mass. Since the ratio i, whose expres- 

 sion we seek, is of no dimensions in mass, it follows at once that D and D' occur only under the 

 form D: D', which is a simple number and may therefore be omitted. It remains to find how 

 i varies with T, r, X, b. 



"Now, of these quantities, b is the only one depending on time; and therefore, as i is of no 

 dimensions in time, b cannot occur in its expression. We are left, then, with T, r, and X; and 

 from what we know of the dynamics of the question, we may be sure that i varies directly as 

 T and inversely as r, and must therefore be proportional to 1 + XV, T being of three dimensions 

 in space. In passing from one part of the spectrum to another X is the only quantity which 

 varies, and we have the important law: 



" When light is scattered by particles which are very small compared with any of the wave 

 lengths, the ratio of the amplitudes of the vibrations of the scattered and incident light varies 

 inversely as the square of the wave length, and the intensity of the lights themselves as the 

 inverse fourth power." 



The dimensional and conversion-factor formulae for the more commonly 

 occurring derived units will now be developed. 



GEOMETRICAL AND MECHANICAL UNITS 



Area is referred to a unit square whose side is the unit of length. The area of 

 a surface is expressed as 



S = CL 2 , 



where the constant C depends on the contour of the surface and L is a linear 

 dimension. If the surface is a square and L the length of, a side, C is unity; 

 if a circle and L its diameter, C is 71-/4. The dimensional formula is therefore 

 [L 2 ] and the conversion factor [7 2 ]. (Since the conversion factors are always of 

 the same dimensions as the dimensional formulae they will be omitted in the 

 subsequent discussions. A table of them will be found on page 3.) 



Volume is referred to a unit cube whose edge is the unit of length. The volume 

 of a body is expressed as 



V = CU. 



The constant C depends on the shape of the bounding surfaces. The dimen- 

 sional formula is [_L 3 ~\- 



Density is the quantity of matter per unit volume. The dimensional formula 

 is [M/ V] or [MX" 3 ]. 



Ex. — The density of a body is 150 pd. per cu. ft.: required the density in grains per cu. in. 

 Here m, the number of grains in a pd., = 7000; /, the number of in. in a ft., = 12; mH = 7000/12^ 

 = 4.051. The density is 150 X 4.051 = 607.6 grains/cu. in. 



The specific gravity of a body is the ratio of a density to the density of a standard 

 substance. The dimensional formula and conversion factor are both unity. 



1 Philos. Mag., (4) 41, p. 107, 1871. See also Robertson, Dimensional analysis, Gen. Elec. 

 Rev., 33, 207, 1930. 



