498 Mr. T. Chaundy: Method of Line- Coordinates for 

 These six quantities clearly satisfy the two identities 



+ m? + ?i 2 =l, 



al+bm + cn 



:a <•> 



In this way the six degrees of freedom represented by the 

 six coordinates are reduced to four, corresponding to the 

 four degrees of freedom known to be possessed by a straight 

 line in space of three dimensions. Considerations of 

 symmetry, however, render it in general more convenient 

 to employ these six coordinates obedient to two identities 

 than to employ four independent coordinates. 



The quantities a, h, c may be given the following physical 

 meaning, if it is desired to visualise them. Tf unit force be 

 supposed to act along the line, then a, b, c, respectively, are 

 the turning moments of the force about the three axes of 

 coordinates, at the same time that, of course, I, w, n, are the 

 components of the force along the axes. It is thus some- 

 times convenient to speak of a, 6, c as the " moments " of 

 the line about the axes. 



(2) Refraction at a single surface. 



Suppose a ray whose direction cosines are (I, m, n) to be 

 incident at a point of a refracting surface at which the 

 normal (drawn into the second medium) has direction cosines 

 (L, M, N) : then we shall require formulae giving the direction 

 cosines (l\ m', n f ) of the refracted ray. 



The first law of refraction, namely, that the refracted ray 

 is in the plane of the incident ray and the normal, is expressed 

 analytically by the fact that I', m', n' can be written in the 

 form 



l'=Al +BL,1 



?n' = A;?i + BM, V (4) 



n' = An+BN, ) 



for some values of the multipliers A, B. 



To express analytically the second law of refraction, 

 namely, that sin (angle of incidence) =/jl sin (angle of re- 

 fraction), suppose that the incident and refracted rays make 

 angles 0, 0' with the normal and that L', M/, N' are the 

 direction cosines of a line in the plane of incidence normal 

 to (L, M, N), e. g., the section by the plane of incidence of 

 the tangent plane of the refracting surface. 



The various directions are then related as marginally 

 indicated. 



