538 PROFESSOR KELLAND ON THE THEORY OB' WAVES. 



also equation (IV.) gives 



kg + rq = (i (4'); 



and equation (V) 



k/+pr=Q (5'); 



If k and r are not each equal to zero, we obtain by equations (4') and (5') 



■L=L 



9 1 

 and since by (2') fg = -p <,, 



... f=-f...=-/ 



y=0,y=0. 

 But, in this, case, by (!'), 



/=0,/-=0. 

 Hence our functions are reduced to 



Y = k , ^^=/- . 



This solution is one with which we have no concern ; it belongs to a state 

 of rest, or of uniform motion. If, however, one of the quantities, as k. is equal 

 to zero, then rp=i) and rq=0\ either, therefore, r=0 or p=Q, g-Q. But if /*=0, 

 9=0 we have by (T) /--g-" , and by (2') fg=0: 



■■■ /=0,g = 0; 



and we are reduced to the same state as before. 



But if ^=0, the only equations to be satisfied are (1') and (2'), which are 



fg+P9=0: 

 and thus we reduce our equations to the form 



F (=foos act +g sin act 

 -\, I =p cos a.c t + (/ am ac t 

 ft^ —a, c {^g cos o c t—f sin act) 

 (f> t— —ac{qcoaac t—psm.ac t) 



39. Now, when ax= — -^ and /=0, our assumed conditions are, that m=0, 



t>=0 ; and since it has been proved that H=0, it follows that ^ t must be equal 

 to when <=0 ; hence we must have y=0: wherefore we require to have either 

 /=0 or^=0, in order to satisfy equation (2'). 



But F2 + 4= = a constant 



hence if cos ac t +g sin ac if +p^ cos'' ac t= const 



2 2 



consequently p'^^a^-P 



and p* is a positive quantity, therefore ^ cannot be equal to 0, hence f= ; 



and p='=^g-—g suppose. 



