On the Intensity of Light, &c, 15 



show this it will be necessary to lay down the following general 

 rule for compounding rectilinear vibrations having the same 

 period, whatever be the difference of their origin and direc- 

 tion : 



Let AA fmd.BB' (Fig. 7), bisecting each other at 0, represent 

 in extent and direction the vibrations 

 to be compounded, and suppose C 

 and D to be two simultaneous posi- 

 tions of the moving molecule, which 

 it would have in virtue of each vi- 

 bration singly. Complete the paral- 

 lelograms OE and OP, and through 

 P describe an ellipse having for 

 its centre, and touching the sides of 

 the parallelogram OE ; this ellipse will represent the resulting 

 vibration; it will have the same period as the compound one, 

 and equal areas will be described in equal times about its centre. 



To apply this construction to the case proposed, it is neces- 

 sary to show that when OA and OB are constant and at right 

 angles to each other, the intensity of the elliptic vibration, or 

 the sum of the squares of the semiaxes, is independent of the 

 difference of origin, or of the position of the points C and D. 



Now in an ellipse, when a perpendicular from the centre on a 

 tangent makes an angle with the major axis, the square of its 

 length is equal to a 2 cos 2 <f> + b 2 sin 2 0. If be the angle which 

 the major axis makes with OA, it will make its complement 

 with OB, and we shall have 



OA 2 = a? cos 2 + b z sin 2 0, 



OB* = c? sin 2 $ + b* cos 2 0, 

 and therefore 



OA* + OB 2 = a* + b\ 



Hence the intensity is independent of the difference of origin, 

 and therefore the rays do not interfere. 



This remarkable circumstance is commonly accounted for by 



