538 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



also equation (IV.) gives 



kg + rq = Q {41); 



and equation (V) 



kf+pr=Q (5'); 



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



1=1- 



9 1 



and since by (2') fg= -pq, 



3 q 

 But, in this^case, by (I'), 

 Hence our functions are reduced to 



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 rjc»=0 and ry=0; either, therefore, r=Oor />=0,y=0. But if /)=0, 

 y=0 we have by (!') f-^g"" , and by (2') fg=0: 



.-. /=0,g = 0; 



and we are reduced to the same state as before. 



But ifr=0, the only equations to be satisfied are (1') and (20, which are 



fg+pq = 0; 



and thus we reduce our equations to the form 



F i=f cos net +g sin act 



y\/ 1 = /J COS ac ( + q sin act 



/t= —uc(g cos a c t—f sin act) 



<p t=. —ac{q cos ac t—psm a c t) , 



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



«=0 ; and since it has been proved that H=0, it foUows that (p t must be equal 

 to when ^=0 ; hence we must have §^=0 : wherefore we require to have either 

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



But F^ + -v)/^ = a constant 



hence (/cos « c / +,^ sin « c t)'^ -\-p'^ cos^ act= const 



P-9' . P' _n 

 2 2 



consequently p^=9^—/"' 



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



and p=^^9=—9 suppose. 



