PROFESSOR KELLAND ON THE THEORY OF WAVES. 107 



that of a be is 2SS 3 (3sin0cos3 



+ 2 D D 3 (3 cos 6 sin 3 6 + k + sin 6 cos 3 6 + k~) 



= -2 (S 4 + S 2 ) (3 sin cos 3 6 + k + cos 6 sin 3 6 + k~) 

 + 2 (S 4 - S 2 ) (3 cos 6 sin 3 + A + sin 6 cos 3 + K) 



= 6 S 4 sin 20 + A-6 S 2 sin40 + *-2 S 4 sin 20 + #-2 S 2 sin 40 + A 

 =4 S 4 sin (2 + *)-8 S 2 sin (4 6 + k~) ; 

 and, lastly, that of a W is 



-2 D sin (S D 3 sin 6 cos 3 6 + k + S 3 D cos 6 sin 3 + )-6 S D D 3 sin cos (9 sin 3 6 + k 



+ 2 S 2 S 3 sin 6 cos 6 sin 3 6 + k-2 S D D 3 cos 2 6 cos 3 6 + k 



-(S 5 -S) 2 sin 2 6 cos 3 + k-2 (S 5 + S-2 S 3 ) sin 6 cos 6 sin 3 + A; 



-6 (S 5 -S) sin cos0 sin 3 6 + k+ 2 (S 5 + S + 2 S 3 ) sin ^ cos 6 sin 3 6 + k 

 -2 (S 5 -S) cos 2 6 cos 



+ 3 (S 5 +2 S 3 + S) sin 2 6 cos30 + k-3 (S 5 -2 S 3 + S) cos 2 B cos 3 + k 



= _ 2 (S s - S) cos 3 6 + k- (6 S 5 - 8 S 3 - 6 S) sin 6 cos sin 3 + k 



Hence, collecting all the terms, we get 

 9 - (b D cos 6 + a D 3 cos 3 + )-46 2 c sin2 d + {(2 S-S 3 ) cos 0-S cos 3 6} 6 s 



-a& 2 {2 (S 5 -3 S 3 + S) cos 3 + A + (3 S 5 -2 S 3 ) cos d + k + (2 S 3 + 3 S) cos 5 



+6 c 2 S cos + 3 a e 2 S 3 cos 3 

 53. If in this expression we write 6=0, we obtain 



&D- -D 8 cos *-(S 3 -S) 6 3 + 4 a b c sink (S 4 -2 S 2 ) 

 o. g . 



If we write =-?r, and dash the sums and differences 



3 ' cos *-(S,'- S') 6 3 -4 a 6c {Si'-2 S 2 '} sin &-&c. =0 





