PROFESSOR KELLAND ON THE THEORY OF WAVES. 537 



And the equation /F + (p-i^=0 gives us 



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

 + (p cos . + 9 sin . + r) {in p — acq cos .-^ mq-\-acpsa\ .+mr)=.^. 



Equating coefficients we get 



/{mf- a eg) +g {mg + a ef) +p {mp -acq) 

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



f{mf~ aeg)-g {mg + acf)+p{mp-aeq) 

 — q{mq + acp) = (2) 



f{mg + ac/)+g {mf- a eg) 

 +p{mq + acp)4-q(mp — acq) = (3) - 



mkf-vk{mf—aeg) +mr p + r{mp — a,cq) = (4) 



mkg + k{mg + acf) + mrq + r{mq + acp) = Q (5) 



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



/{m/—aeg)+p{mp — acq) + m{k'' + r')=0 (I;) 



From equation (2) alone we get 



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



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



From (4) 2m{k/+rp) = ae{kg + rq) (IV.) 



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



If it be allowable to eliminate f'+p'-g'-q' between II. and III., we obtain 



m _ ac 

 2a c 2 m 



or w" + a'c^=0 



»j=0 , a c = ; 



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

 it is obvious this case is that in which Y 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 f+p^—g^'—q^ or fg+pq is 

 equal to zero. 



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

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



/g+pq=0 {2) • 

 But the first equation gives m (/» + / + A' + ^) = by means of (2') ; 

 which again, combined with (!'), gives 



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



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 



