REFLECTED AT THE SURFACE OF A CRYSTAL. 41 



We proceed next to find the variations in a, ft 7, a,, p a 7,, due to a variation 

 in the co-ordinates at, y, z. Let a + 8a, &c. be the values of a and of the other 

 quantities, when x + 8 x, y + 8 y, z + 8 z are the values of the co-ordinates. 

 Now da = (dl-d'K)sin(p + dR l , &c.=&c. 



Therefore we must commence by finding the values of 8 I, 8 R, and o R,. 



Now if p, j/, p", p'" be the perpendiculars from the origin (supposed above the 

 plane of yz and of each of the waves) on the incident, reflected and two refracted 

 waves, it is evident that p =x cos<p+y sin <p + const. 



p' =x' cos (f> y sin (p + const. 



p" x" cos <p,+y" sin (f), + const. 



p'" = x!" cos </>' + /' sin <j>' + const. 



and therefore I = a cos ( -=- p + ct. J 



= acos 



f -= . x cos <p+y sin (p + ct -f const. J 



(O ,r- \ 



r p' + ct + const. \ 



= 6 cos I r a? cos + y sin (p + ct + const. ) 

 T = c cos / . jf cos (j), +y" sin ^ + ct + const. J 



\ '*/ / 



(O _. _ v 



_ . a/"cos<^)'+y"sin0' + ct + const. J 



where it is to be observed that, as the reflected pencil moves in the opposite di- 

 rection to the incident, the sign of j/ within the circular function is negative. 



We will abbreviate these equations by denoting -^ cos (p by e, -^-sin by/ 



&c., and suppressing the accents to the co-ordinates ; thus we shall have 



I = a cos (e x +fy + ct), I' = a' cos (ex +fy + ct), 



R =6 cos( ex+fy + ct + fi), R'^cos ( ex+fy + ct + k"), (3.) 



R,= A e~ Jmx cos (fy + ct + f), T = c cos (e, x +fy + ct -f m), 



T=c f cos (e' x +fy + ct + ), T,= C ? "V cos (fy + ct +JB). 



It must be remarked that f is the same in all the equations, since !^r = 



sin d> sin d>' \ v. sin d>. 



TT- 5rr-, from the circumstance that V=~= ^- 

 A, A A sin 



Now 8l = acos(ex + edx +fy +/8y + ct') a cos (e x +fy + ct) 



= I (1 cosedx+f8 y) a sin (ea;+/y + ct) sin (ex+fdy) 



-- 



& uX 



and in the same way the other increments are easily determined. 



VOL. XV. PART I. 



