12 Mr. Tovey's Researches in the Undidatory Theory: 



taking, for the first equation, the values of A t) and A ? which 

 involve the arc {nt — kx), and for the second equation, the 

 values involving \n t—kx—b). We shall then have 

 («« + 5H-ps'2) sin {nt-k x)—(s,-\- p s'^ cos{nt-k x) = 0, 

 (p[n^^^)-\-s^)sm{nt''kx—h)-{ps^^\■s.^)cos{nt — kx-'b)=0\ 



and since these equations are true for all values of / and x^ 

 they resolve themselves into 



71^ + 5 + ^52 = 0, 



p(w^+s')+52= 0, ,, V 



^5/ + .93 = 0. 



These equations may be satisfied by means of the four ar- 

 bitrary quantities n, p,k,b\ the last two being contained in 

 them implicitly. 



From the first and second of these equations we derive 



Let .,.^ 



4 



then the two values of n^ will be 



n^ =z — s— s, W/« = - 5' + « ; (5.) 



and those of p 



f. = ^. P.= -^- (6-) 



These values being substituted in the assumed expressions 

 for >) and ?, we have 



>j = a^ sin (n^ t—k x) + a^^ sin {n t—k x) , .^ x 



i^^=p^a^sm{n^t — kx—b)+p^Ja^^{nt — kx^—b), ^ '^ 



It has been seen (at p. 501 of your 8th vol.) that any func- 

 tion of X and t may be expressed by a series of which each 

 term has the form {psmk x -^-q coskx) ; where p and q are 

 functions of t, and k a constant quantity. P'or the same rea- 

 son any function of t may be expressed by a series, each term 

 having the form A sin w ^ + B cos n t^ where t is the only 

 variable. Therefore, iot [p ^m k x-^q cos k x) we may write 

 ( A sin w ^ + B cos w /) sin A: or + { A sin w ^ + B cos n t) cos k x, 

 which, by the rules of trigonometry, may be reduced to the 



^o™ asin{nt--kx'-b) + a'sm{nt-^kx'-b^); 



