338 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



investigation the same degree of accuracy as belongs to the hypotheses, because we have no where 

 infringed those hypotheses by analytical approximations. It is easy to shew that we cannot regard 

 our second hypothesis as being strictly correct. For if it were a possible hypothesis, then as the 

 first cannot be at the same time true, the quantity denoted by ri'- in equation (9) must be regarded 

 as a slowly varying function of t. The equation for p then assumes the form f = F{x,t) - gy 

 + in'y^; which involves the same impossibility as before, because at any given moment, at the 

 junction of the wave with the quiescent fluid, the pressure depends on y^ as well as on y, which 

 cannot be the case. Hence our second hypothesis is certainly not mathematically correct, u must 

 therefore depend on y as well as on x. 

 We come now to the consideration 



OF THE NEGATIVE SOLITARY WAVE. 



Iv this case, we are to represent the constant of equation (7) by - n' ; the equations therefore 

 which are peculiar to the wave we are now investigating are, 



-91^= dfd^u + %i,d^u — {d^u)' (8'), 



S,''y = n-y (9'). 



From the last we obtain 



y = ^(e°'-"± 6-"' + °), 



and .-. ^,2/ = ^w(e"'-''=F e""'""")- 



Now by the nature of the case S,y = when the particle has gained its lowest position ; 

 but S,y can never become = 0, unless we use the upper sign ; the upper sign must therefore be 

 used ; and consequently we obtain 



y = ^(6"'-" + 6-"' + ") (10'), 



V = S,y = n^Ce"'-"- «-'" + '■) (Ii'). 



V g»(-a_g-„l + a 



Also rf."=- — ->^- ,a-.^,-..., 02). 



The form of (9') shews that the force which regulates the vertical motion of each particle act.s 

 upwards, and consequently if the particle oscillate (which it must do if it be part of a wave) its 

 motion at first inust be downwards ; it then comes to a minimum altitude above the bottom of the 

 canal and then rises again to its original level. Let h be as before, and k the altitude of the lowest 

 point of a wave above the bottom of the canal ; then proceeding as in the corresponding part of 

 the investigation for the positive wave, we obtain 



««jt - a = (13'), 



/c = 2 J, 



k f 

 h= (i 



fit/, ~ nik ntk~nth\ 



e + e 



Tax 



) (1*); 



1 /tt 



and .-. t^- t, = - log„ tan (^- + i cos " ' 



Since v is negative in the fore part of the wave, and positive in the hinder part, nt is less 

 than a as long the particle is situated in the fore part, greater than a when it is in the hinder part, 

 and equal to a when the vertex of the wave passes it. 



