108 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



4 a b c {S 4 -2 S 2 + S/-2 S 2 '} sin k 

 = b (D - D') + I a cos k (D 3 - D' 3 ) 



+ a b 2 cos* (5 Sg-6 S 3 + 5 Si-5 S 5 '-6 S 3 ' + 5 S')-6 c 2 S- S'-3a c 2 cos * S 3 - S 3 ' 

 or 4a6c{2S 4 -4S 2 }sinA 



= b . a S D 3 sin A + &c. + Sec. - b c -^ D D 3 sin A- &c. 

 a oe D c 



if sin k is not equal to ; then 



2S 2 )=^ .? SDi.- 

 u o 



3D 



3(S 4 -2S 2 )D=D 3 . 

 or, if we choose to retain all the terms to the 2d order 





If we write 6=^ and accent the letters 





_ I b (D - D') - 1 a cos k (D 3 - D 3 ') - 6 3 (S 3 - S - S 3 ' + S') 



Now z= h + (e z -e- K *) sin 0+-?- ( e 3 "^ - e" 3 " sin (3 + A), 

 c ct (3 c a 



*(h+~ Da sin/fc) (A + -^- Bs sin k) 

 D=c -e 



= D + S aD3sinA; , if D , S o are the values of D and S when we omit k, 



O C 



D-iy= 2S aD3 sm/fc nearly, 



6c 



10D 5 aD 3 



5 O 5= - g^ - SID A 



SEC. &c. 



