406 Tr<m$aetion& of the Society. 



according to which it is always possible to find a quantity A and 

 an angle ft such thai the equations are fulfilled — 



f A ens /3 = A a cos /3j + A 2 cos fa 

 \ A sin /3 = Aj sin & + A 2 sin fa 



and it' we substitute these values in (3) we obtain — 



5. £ = £ x + £ 2 = A cos /3 sin t — A sin /3 cos t = A sin (t — j3) 



This represents a new wave of amplitude A; the value of A 

 can be obtained by squaring equations (4) and adding them together ; 

 for the squaring gives — 



] A 2 cos 2 /3 = A^ cos 2 fa + A 2 2 cos 2 fa + 2 A, Ao cos fa cos fa 

 ) A 2 sin 2 /3 = Ai 2 sin 2 fa + A 2 2 sin 2 £ 2 + 2 A x A 2 sin fa sin fa 



and remembering that sin 2 + cos 2 of any angle is equal to one, the 

 addition gives — 



A 2 = A x 2 + A 2 2 + 2 A x A 2 {cos fa cos fa + sin fa sin /3 2 } 



The terms in brackets represent cos (fti — /3 2 ), hence we get the 

 general solution of our problem — 



6. A = J AS + A 2 2 + 2 Aj A 2 cos (fa - fa) 



and having obtained A from this, we can get the phase-angle from 

 (4), for dividing the second by the first, we obtain — 



A! sin fa + A 2 sin fa ^ 

 tg P ~ Ax cos fa + A 2 cos fa 



It hardly needs stating that equations (6) and (7), being a per- 

 fectly general solution, include all special cases that may occur. 

 That they cannot, however, be applied to light from independent 

 sources has already been laid down, and needs no further mention. 



Equation (6) is identical in form with the one obtained in 

 mechanics for the resultant of two forces ; and as this is a remark- 

 able and sometimes convenient relationship, I will briefly prove it. 



Let F x and F 2 in fig. 7G be the two forces acting at point G, let 

 their direction be defined by the angles /3 X and /3 2 respectively 

 which they form with some fixed direction C Z. Then it is well 

 known that the resultant force corresponds in magnitude and 

 direction to the diagonal C E of the " parallelogram of forces " 

 C I) E F. 



* It may be pointed out that, if A is always, as is usual, given the positive sign, 

 the quadrant in which /3 is to be taken must be determined by the sign of the right 

 hand sides of equations (4) in the usual manner. 



