330 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



Integrating (3) with regard to y, remembering that it and therefore also d^u, is independent 

 of y, we find 



t) = - yd^ii, (5), 



no arbitrary function of x being added to this integral, because manifestly u = when y = 0, 

 whatever be the value of so ; and no function of t is added because from (4) t enters with x only. 

 By means of this result eliminating v from (2) it becomes 



dyP = - g + {didrU + ud^ht - {d^uy\y (6). 



Now d .d,p = d^.dyp; and as appears from (1) d,p being independent of y, d^.d^p = 0, 

 consequently d p must be independent of x ; from which it follows that the coefficient of y in (6) 

 though a function of u is not a function of x, and therefore not of t by (4) ; and of course it is 

 not a function of y, consequently it is constant both with respect to x, y, and t ; 



.-. constant = d^d^u + udj'ii - {d,uy (7). 



Before proceeding farther it is necessary to ascertain whether this constant have a positive 

 or negative sign. We may ascertain this as follows. 



Let us use the letter S as the symbol of differentiation, taking x and y to belong to the same 

 particle through the time St; then it is well known that instead of the equation (2) we may use 

 the following which is exactly equivalent to it, viz. 



dyP = - g - ^?y, 

 which being compared with (6) gives, 



^i^y = - [dtd^u + udjti - (d^ufly, 

 = — (constant) y. 



Hence the force which urges the vertical motion of any particle varies as the distance of the 

 particle from the bottom of the canal, and has always the same sign. Consequently when the 

 original displacement of the fluid is such that any particle attains thereby a higher position than 

 it had when in equilibrium, the above force must act so as to bring it down to its original level ; 

 i.e. the force must then be negative. Hence for what Mr. Russell calls the positive wave the 

 above constant is positive. In a similar way it appears that for the negative wave the constant 

 has a negative sign. It is therefore now necessary to separate our investigation into two branches, 

 treating separately of these two varieties of the solitary wave. 



OF THE POSITIVE SOLITARY WAVE. 



In this case, representing the constant by n^, we have for discussion the equations 



n- = dtd^u + ud/u - (d^ti)' ; (8), 



^'y= -n'y (9) 



which belong only to the variety of wave we are now considering. The latter will furnish us 

 with the law of the vertical motion of each particle ; and it shews that it is expressible in the 

 form of a sine or cosine of an angle the variable part of whose argument is nt. 



