150 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



But, if we introduce the expansions (21) for i/, , we find 



n < n + 1 > + * }I "- P '- ' ' (37) ' 



whence the final equation which replaces (31) will be obtained by replacing C^ by 



Ci + ^{n( + l) + r}n ....... (38) 



in the right-hand member of (28), and h by K in the left-hand member. 

 Thus we obtain 



+ tf 3;+. + s {(* + )( + ) + }!:* . (39); 



and therefore, on separating out the parts of F. due to C^, y', respectively, we find 



'-*-.:- i:.C" +,'Q,, 2 = G: ........ (40), 



where 



2 ) ( + S )+ " 



and 



For the special case s = the equation (40) reduces to the equation (23A) of 

 Parti. 



5. The Period -equation for the Five Oscillations in an Ocean of Uniform Depth. 

 To determine the periods of free oscillation we may proceed exactly as in 6 of 



