Interference Phenomena. 533 



where F(\i, XJ means the function into which F is changed 

 by substituting everywhere X 1 for X 2 . If> therefore, we trans- 

 form a function of one variable t, F(/), into one of two variables 

 t and t f j ¥(t,f), by simply writing t' for t in part of the ex- 

 pression, we may put 



7r] F(t)dt = I I I F(X 1? X 2 ) cos ^(X x — X 2 ) dX l dX 2 dfc. 



J — co J — oo J — cojo 



If we apply this to the product f(t)^>(t) we find 



f(t)cj)(t)dt= I I I /(X 1 )(/>(X 2 )cos/c(X 1 — X 2 )dX 1 dX 2 dfc. 



-co J-^J-^JO 



The theorem expressed by this equation may be written 



7T 



where 



A x = I /(X) cos kX dX, ^i=\ f(X) sin /cXdX, 



J-co J-co 



A 2 =\ 4> (X) cos /cX dX, B 3 =| </>(X)sin/eX<iX. 



J-co J-oo 



The special case in which the two functions / and </> are 

 equal has been proved by Eayleigh (Phil. Mag. xxvii. 1889). 

 The excess of energy of two trains of waves over that of the 

 separate ones is thus seen to depend in a simple way on the 

 analysis of each by Fourier's theorem. If the two trains are 

 of the same type, one following the other at an interval of 

 time 2r, we can put the excess of energy E into the form 



/-» _|_ oo /■» -f oo n CO 



2 E = /( Xl + T )^ X 2 - T ) cos /e(X x - X 2 ) dX x dX 2 die. 



Substituting ^i=X 1 + t, A& 2 =X 3 — t, this becomes 

 2 E = I /(/*il/W cos^O^i — ^ — ^d^d^d/c 



J+cop + cop* 

 ./W/W cos % KT c °s * (/*! -ftj) dfidtodic, 

 -co J — co Jo 



J+co p+co 



f(fji) cos k dfx, B=| /(/u) sin Kfid/jL, 

 -CO J — CO 



f» oo 



|.E= (A 2 + B 2 )cos2/ct^ (5) 



But (A 2 + B 2 )d/e is proportional to that part of the energy 



