ANALYSIS TO THE DYNAMICAL THEORY OP THK TIDES. 147 



Again, on -replacing lf 3 by their expansions in (18), the two members may be 

 expressed as series of associated functions by means of (9), (6). Thus we find 



- * I. (* * I) J P-, = 2^* [^ P.., + il p. .,] , 

 whence, on equating coefficients, we deduce that 



Finally from (20), on expressing the two members by their expansions and 

 equating coefficients, we obtain 



(27) 



The relations (26), (27), enable us to eliminate the auxiliary constants fi',, <_,, 

 < +1 from (25), and thus to express F, as a linear function of C*_ 2 , C*, C?, +2 . On 

 substituting for <_,, < +1 from the formula (2G) in (25), we obtain 



_ ., _ 



-s- n * 



2^T **'* ' " ST+1 K 



- - 



(M + a+ !){( + 2) ( + 3) + <r} {n-s+ 1 M + + 2 1 



(2n + 3) {( + ]) (n + 2) - a} [ 2n + I P * ' 2n + I, ^' +2 J 



_ __ 2 ( - a>( -a-l) {( - 1) (a - 2) 



(2n - 1) (2n - 3) {(n - 1) n - a] 



- - _ Q" _ _ | /? ; 



. - i \/ii /O'ii1\/*)"ii t "*\f/-ii1'WiiO\ ~\ \ " il 



ft "^ J. I It ~~^ u f I _ / ' *T~ J. I ( _ // ~^~ * * / i ( /( i" i )\ ll ~} J ^^ O" f 



2 ( + 8 + 1) ( + a + 2) {( + 2) ( + 3) + o-} 

 (2 + 3) (2n + 5) {(n + 1) ( + 2) - <r} 



and this, by means of (27), gives 



-Ml fo - s) Qt - a - 1) 



< r , 



4o) 8 a'~ (2% - 1)(2 - 3) {( - 1) M - < 



, __ ( + * + 1) (-- + 2) r , 



r (2n + 3) (2 + 5) {( + 1) ( + 2) - <r} 



where 



__ s^ ___ (n - s) Qi + <) {(n - 1) ( - 2) + q] 

 " "" <r {n (n + 1) + o-} (2-n -i) (2n + 1) {(n - 1) - <r} {m (w + 1) + <r] 



(n + s + 1) (M - 8 + 1) {(?t + 2) Q + 3) + <r} 

 ~ (2n + 1) (2 + 3) {( + 1) ( + 2) - <r} { ( + 1) + a}' 

 V 2 



