PROFESSOR KELLAND ON THE THEORY OF WAVES. 537 



And the equation /F + cpy^—O gives us 



(/cos . +^ sin . + k) (mf— a eg cos . + mg + a c/sin . + m A;) 

 + (jocos. + ^sin . + r) {mp — acq cos . -^ m q + a c p mi . + m r) =0. 



Equating coefficients we get 



fimf—acg) +g{mg-\-acf) +p(mp — cccq) 

 + q(mq + acp) + 2mk^ + 2mr' = (1) 



/(mf-acg)-g{mg + acf)+p(mp-acq) 

 — q(mq + acp)=0 (2) 



f{mg + a cf) +g {mf- a eg) 

 +p (m q + a c p) -^ q (mp — ac q) = (3) 



mkf+k{m/—a,eg) +mrp + r(mp — acq) = (4) 



mkg + k(mg + acf) + mrq + r(mq + aep) = (5) 



38. From combining (1) and (2), there arises the following equation : 



/(mf—acg)+p(mp — aeq) + m(k'' + r^)=0 (L) 



From equation (2) alone we get 



m(r + p^-f-q^)=2ac(/g+pq) (II.) 



From (3) 2m(fg+pq) + ac (f+p'-g'~q') = (III.) 



From (4) 2m(kf+rp) = ac(kg + rq) (IV.) 



Prom (5) 2m(kg + rq) + ac(k/+pr) = (V.) 



If it be allowable to eliminate /'+/-/ -9^ between II. and III., we obtain 



m _ ac 

 2ac 2 m 



or m' + a'c' = 



m=^0 , a c = 0; 



and all the equations are satisfied without giving any other conditions. In fact, 

 it is obvious this case is that in which F t, ■^t are constant. 



Now, the only reason which can operate to prevent this elimination being ef- 

 fected, is the circumstance that one of the quantities /'+/—/—?' or fg+pq is 

 equal to zero. 



And, by means of the same equations, it appears that both the quantities must 

 equal zero; or /'+/-/-?' =0 ^1') 



/g+pq=0 (2) ' 



But the first equation gives m (/' + p' + A;' + r") = by means of (2') ; 

 which again, combined with (V), gives 



m{g' + q' + k' + r') = {) (3') 



Either therefore m = 0, or g, q, k, r,f, p, are all separately equal to ; but the 

 latter condition cannot be true, 



VOL. XIV. PART II. 4 Z 



