Application of Undvlatory Theory. Bij A. E. Conrady. 405 



equal to the length G — D. To obtain the resultant of the two we 

 have merely to add the two displacements which become super- 

 posed in each point. 



On the line E — H 1 we see that wave X has a displacement E F 

 in the positive sense, whilst wave Y has a displacement G H in 

 the negative sense ; hence we obtain the resultant displacement 

 by marking off F 1 E 1 = E F, and then from E 1 going back by the 

 amount E 1 H 1 = G H. The resultant displacement here is there- 

 fore = F 1 H 1 . 



In the position I — M 1 the result is different, for here both 

 combining waves have displacements in the same negative sense, 

 hence we get a large resulting displacement = I 1 M 1 . 



By carrying out this process in a sufficient number of points, 

 we get the result of the combination in the form of a new wave 

 (X -j- Y), which differs in phase and in amplitude from the com- 

 bining portions, but retains the same wave-length. 



Mathematically, the solution is arrived at by bearing in mind 

 that in our wave-equation (I) a difference of phase is expressed by 

 a change in the value of X. If we solve the bracket in (I) we get — 



= A sm < — V t X \ 



\ \ x i 



For simplicity's sake we will introduce simple symbols for the 

 two parts. The first contains the time t and is an ever-growing- 

 angle ; in my paper of November 1904 I called it a, but in order 

 to make it easier to remember that this angle involves the time, I 

 will now and henceforth call it t ; the second angle is the difference 

 of phase compared with that at some fixed distance from the source 

 of light, and 1 will retain the symbol fi for this. We may thus 

 write two combining wave-motions as — 



'8 





& = Ai sin (t - fr) 

 A 2 sin (t - (3 2 ) 



and we can combine these by solving the sines ; we obtain — 



C fi = Ax sin t cos /3 X - Ax cos t sin & 

 *• \ £ 2 = A 2 sin t cos /3 2 - A 2 cos t cos #, 



and these give the sum — 



3. £ + f 2 = sin t (A x cos /3 X + A 2 cos /3 2 ) - cos t 

 (Ax sin @i + A 2 sin /3 2 ) 



All the quantities in brackets are independent of the time ; we 

 can simplify them by utilising a general trigonometrical theorem, 



