428 Mr. Tovey on the Vndulaiojy Theori/ of Light. 



the supposition, be insensible; hence we shall have, by sub 

 stituting for A x,^ A sf its value previously found, 



v,,^ = cos^ 9 . "'- S . 4/ (?•) (A x'- A 1/"- sin^ fl' + A^^A z<^ cos^ 0') 

 + sin-' 9. -|-2.rP(/-)A.^'^Ay^; 



or, v,;^ = c' cos= sin^ 9' + c" cos^ 9 cos'' fl' + c"' sin^ fl ; 

 if, for the sake of abridgement, we put 



-- 2 . -^ (;•) A .r'^ Aj/'^ = c', -^ 2 . ^ (r) A .r'^ A z'-^ = c'^ , 



2..^(r)Ay^A 



^/■j _ 



( 1 7.) Let O j' (fig. 3,) be the common axis of ^•' and x, ; Oy, 



O^', Oy,0<, O^r, Oy, o . 



tlie axes of y\z' , y^., z, , x,y, z; 

 y' O j/, = fl' , and x' O x = 9, Now 

 O.T being, by the supposition of 

 art. 6, in the plane of x' Oy,, we 

 will suppose Oy to be also in the 

 same plane. Then y/Oy = x' O x 

 = 9 , cos x' Oy = sin 9, cosy Oy 

 = cos 9, cos 9' , cos ;;' Oy = cos 9 

 sin 9' ; and thus, by the last expres- 

 sion for 7'/ , we arrive at 



= c- COS' ;:' Oy + c'- cos" j/' Oy + c"' c 



In deducing this equation we have supposed Ox', Ox, 

 Oy, to be in the same plane; but we take for granted that 

 the value of v"- would not be sensibly affected by turning the 

 coordinates upon the axis of j/; because 

 m 



v.^ = 



4' (?) A .r'' A :;-, and this sum, being symmetri- 

 cal with respect to x and ;;, retains the same value when x 

 and z are interchanged. Hence it follows that the equation 

 is true in general, and, consequently, that if we change the 

 angles z' Oy, y' O y, x' Oy, for z' O ^, y O z, x' O z, it gives 

 also the value oft'/-. 



(18.) It will be remembered that t';, is the velocity of waves 

 movino' in the direction of Ox, and consisting of vibrations 

 in the direction of O - ; and that v' is the velocity of waves 

 niovinfT in the same direction, and consisting of vibrations in 

 the direction of Oy. The directions of O^ and O -, which 

 determine those oi the vibrations must (art. 15.) be so taken 



