116 Sir W. Thomson on the Front and Rear of 



type-solution of (4) and (13) for free waves, 



1 



■ gl2 



(&+# + *#)* 



e A{b+y + ,x) (14) ? 



where i denotes \/ — 1. Whence, as a real solution by adding 

 the values of (14) for c and —i, and dividing by */2, 



where r=[(y + b) 2 + a:*f J 



Curves representing calculated results of this solution for 

 free waves were shown at the meeting of the British Associa- 

 tion (Section A) at Birmingham in September, and at the 

 last meeting (December 20) of the Royal Society of Edin- 

 burgh. To build up of it a solution for a uniformly maintained 

 procession of waves (a double procession it shall be, of equal 

 and similar waves travelling in the two directions from # = 0) 

 take 



P(0=j"«a*(t) (16); 



and 



F= - ( *dt sin &>*' P(*-*')= - ( *dt sin a(t—f) P(0 . . . (17). 



J o •) 



Since <f)(t), as we have seen, satisfies (13), P(£) must satisfy it 

 also. Hence 



dB(t) d*V(t) 



for all values of oc, y, and t. Now by differentiation of (17) 

 we find, because P(0) = 0, and by (16j, 



^ = - f W sin cot f j?(t-t f ) = - jVsina^(*-V) . . .(19); 

 and differentiating this, we find, because <£(0) = (r -\-y + 6) V -1 , 

 g = - fe±f^!« -- j"tf sin * * «,_ 



= _(r+y+6) t rinwf _j , ^ 8ina)(( _ jgg0 -w 



