PROFESSOR KELLAND ON THE THEORY OP WAVES. 

 u = b (e"y + e-"y) sin 6 + a (e 3 ^ + e ~ 3a 2') sin (3 6 +*) 



cos - -.* cos 



then W-(^ 



By substituting these values of u, v, $ when yz, i. e. of ,,,, 0, in equation 

 (11) and writing S 3 for 2a ~+<e~ : "*, &c. we get after dividing by a; 



{& S 2 sin 2 + 2 6a S S 3 sin 0sin (3 6 + A)-6 2 D 2 cos 2 0-2 6a D D 3 cos cos (30 



x { 6 S cos 6 + 3 a S 3 cos (3 + A) } 

 cos0 + 3ac 2 S 8 cos(30 + #)=0, omitting a 2 6. 

 That is, 

 9 {b D cos 6 + a D 3 cos (3 6 + *)} 



- 2 ,c{4 6 2 sin cos + S S 3 a b (3 sin cos 3 6 + k + cos 6 sin 3 + A) 



- D D 8 a 6 (3 cos sin 3 6 + k + sin 6 cos 3 6 + A) + 12 a 2 sin 3 + k cos 3 + k} 



- 2 b 3 S D 2 sin 2 cos - 2 a 6 2 D sin 6 (S D 3 sin 6 cos 3 + k + S 3 D cos 6 sin 3 + A) 

 -6 a 6 2 S D P s sin 6 cos sin (3 6 + k} + 6" S 3 sin 2 cos 0-6 3 D 2 S cos 3 B 



- 3 6 2 a D 2 S 3 cos 2 6 cos 30 + /fc + 6 c 2 S cos 6 + 3 a c 2 S 3 cos 3 6+~k = 0. 



52. In this expression, we will seek out the different coefficients of ft 3 , ft 3 a, &c. 

 and then collect them into one sum. 

 The coefficient of tf is - 4 c sin 2 ; 



that of 6 s is - 2 S D 2 sin 2 6 cos 6 + S 3 sin 2 cos 6 - D 2 S cos 3 Q 



= S sin 2 Q cos (S 2 - D 2 ) - D 2 S cos 6 (cos 2 6 + sin 2 6) 



= 4 S sin 2 cos 0-D 2 S cos 



=2 S sin 2 sin 6- D 2 S cos 6 



= S(cos0-cos30)-D 2 S.cos0 



?=Scos0 S cos 30 S 3 cos + S cos0 



= (2 S - S 3 ) eos 6 - S cos 3 6 ; 



