Oscillations of a Canal of Circular Section. 327 



Using (4) we find that 



^=Q + Xr^ n (Pnsinn6>-- P ^cos^). 

 r i g n y 



In order to satisfy (2), ^ must be constant for r= a and 



7r<6<2ir. We may, without loss of generality, assume 



that o/r — Q = l. Then writing a n P„=S ;i and a 2 a/g = x, we 



have 



1 = Si sin + S 2 sin 20 



-*(S o cos0+ |^cos20 + | 2 cos3<9 ). . (5) 



I£ the constants x, S , S l5 S 2 , &c. be chosen so that this 

 equation holds for all values of 6 between ir and 27r, then 

 </> satisfies all the prescribed conditions. 



Since the series on the right of equation (5) must be 

 symmetrical about OY, S = S 2 = S 4 =0. The cal- 

 culation of the other constants is extremely laborious, and 

 we shall retain only the first three. To solve for ,r, Si, S 2 , 

 and S 3 , we require four equations. Since the series is 

 symmetrical about OY, it is necessary to equate it to unity 



only for the range w<0< — . Hence, putting = 7r, 



77" 7j- 37T . " J . 



7r-f -r, 7T+ -:;, and -^- in succession, we obtain from (5) 





»— (M 



■+»> 





i=-|-s s - 



S 5 /S, S3 S; 



2 '''I -1 " 8 (5 



)■ 





, s^a.i 



^5\/3 / Sj S; 



♦*> 



2 + 



2 -' l V~ 4 ~ 8 



1= — Si + 83 — 



3 •/ Sl o.^> S -A 



iminating Si, S 3 , S 5 , 









*• ;• 



X X 



4' 6 



= 0. 





1. i+j, 



1_? l_f 



8' 2 6 







v/3 £ 



*/3 , x 







5 2 4' 



8' 2 







1, i-|, - 



x .. x 

 4 b 



• • 



(G) 



(7) 



