m 



422 Mb. OBRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



When the vibrations are transversal and the waves plane, the part 



of the equation (B) under the sign 2 reduces to the form mB-^, 



as appears from Articles (20) and (22) ; hence, if we put for a the assumed 

 value a cos k(vt — u), it appears that the part of the equation (B) under 

 the sign 2 reduces to the form - m Bk*a cos Jc (vt - u). 



To reduce the part vinder the sign S put for a moment 



a, — a (or A a) = a, — a 2 + a 2 - a, where a 2 = a, COS k (vt — w), 



= Aa 2 + a 2 — a, and similar expressions for A/3 A 7, 

 and it becomes (since a 2 — a, /3 8 - /3, 72—7 may be brought outside 2 ) 



;£(» -a) + 2,»i,{0(r') Aa 2 + - <£'(r')A#(A#Aa 2 + AyA/3 2 + AxA7 2 )>, 



the part of this expression under the sign 2 is exactly similar to that 

 we have just reduced, having 2 /9 m lf <j>, r' , Ax, Ay, Ass, Aa 2 , A/3 2 , A 72, in- 

 stead of 2, m,f, r, hx, §y, Sz, 8a, 5/3, §y respectively; hence it must in the 

 same manner reduce to the form 



— m^' k 2 a t cos k (vt u), 

 where B' = 1 2, U (/) Ax* + -, (p'ir^AtfAfl . 



Hence, when we have substituted the assumed values of a, /3, 7, a ti 

 (S t , y t , in the equation B, supposing the vibration transversal, we obtain 

 this result [dividing out cos k(vt — u)] viz: 



— &Va = - mBk*a + m l C(a i — a) —m l B'k' i a i (1); 



and if we substitute in the same manner in the equations for -~ and 



d 2 y 



-gny-, we obtain precisely the same result. 



Treating the equations {B) in a similar manner we obtain a similar 

 result, namely, 



- &Va,= - m t B t k 2 a t + mC (a - «,) -mB'tfa (2), 



where B, = | 2 {^ (r^xf + £ f (fj **,%/! . 



