[h^m^ - L (m + m^)] V + le + (I^ + h^m^ - h^Lm^) e^ = [ik] 



This last equation can be integrated at once as follows: 



[h m - L (m + m ) ] V + le + (I + h^m - h Lm ) 9 = a + 6t 

 oo o oooooo 



where a and 6 are arbitrary constants and t denotes the time. A 

 corresponding special solution of [73a, b,c] exists if c =0, namely: 



e=eQ = Y; v=6+ yst 



where y and 6 are constants which are easily expressible in terms of a 

 and 3- This solution represents a steady lateral motion in which the 

 foil does not disturb the stream. By means of the integrated equation, 

 it is possible to eliminate and thereby reduce the equations of motion 

 from three to two. It seems simpler, however, to work with all three 

 equations . 



In looking for harmonic motion, it is apparently less complicated to 

 use as equations of motion Equations [T3a] and [T3c], with signs reversed, 

 and Equation [7^]. Assuming that in these equations v, 6, and 6^ are 

 proportional to e""" , then canceling e , and also dividing Equation [7^] 

 by -0) , gives the following result, provided o) ^ 0: 



* The foil does not disturb the stream because v = 6 represents a steady 

 displacement with 6=0, whereas Q = y and v = ySt represent a steady 

 motion in which the foil slips chordwise through the water thus: 



V = yS " 



i 



ELS seen by observer as seen by water 



83 



