442 Mh STOKES, ON THE STEADY MOTION 



The integral of this equation may be put under different forms. By 

 integrating according to the general method, we get 



U = F(x + \Z~^ly) + f(x - V~\y). 



Now it will be easily seen that U must be wholly real for all values 

 of x and y, at least within certain limits. But F(a) may be put 

 under the form F x (a) + s/~^\ F 2 (a), where F x (a) and F» (a) are wholly 

 real. Making this substitution in the value of U, we get a result, 

 which, without losing generality, may be put under the form 



U = F(x + y/^ly) +F(x - \/^ly) 



+ v^T \f(x + s/-Tl|0 -f(x - n/^IjOI. 

 changing the functions. 



If we develope these functions in series ascending according to integral 

 powers of x, by Taylor's Theorem, which can always be done as long 

 as the origin is arbitrary, we get a series which I shall write for 

 shortness, 



u = 2cos [r y *) ^(y) - 2 sin (a$ x )f^> 



the same result as if we had integrated at once by series by Maclaurin's 

 Theorem. 



It has been proved that the general integral of (5) may be put under 

 the form 



where a* + /3 s = 0. Consequently a and /3 must be, one real, the other 

 imaginary, or both partly real and partly imaginary. Putting then 

 a = c^ + y/ — Ifbi = /3, + s/ — 1 /3 8 , introducing the condition that 

 a + /3' 2 = 0, and replacing imaginary exponentials by sines and cOsines, 

 we find that the most general value of U is of the form 



U = 2^ 6 " (c0Sl ' x - sin w +o) . cos n (sin yx + cos yy + b), 



where A, n, y, a and b have any real values, the value of U being 

 supposed to be real. 



