274 Sir W. Thomson on a Broad River 



using in (67) the expressions so obtained, we find 



V=co%(y)+c l % 1 (y)+c 2 %(y)+c 3 %(y) . . (70), 



where c , c ly c 2 , c 3 are four arbitrary constants, and § , $ 1? § 2 , 

 $ 3 four functions, each wholly determinate, expressed in a 

 series of ascending powers of y which by (68) we see to 

 be convergent for all values of y, unless fi be zero. The 

 essential convergency of these series proves (as in § 39 for 

 the case of no gravity) that the steady motion (w = 0, v = 0, 

 w = 0) is stable, however small be fi, provided it is not zero. 



44. The less is fi, the less the convergence. When /j, is 

 very small there is divergence for many terms, but ultimate 

 convergence. 



45. In the case of /^ = 0, the differential equation (QQ), or 

 (67), becomes reduced from the 4th to the 2nd order, and 

 may be written as follows : — 



This, for the case of two-dimensional motion (^=0), agrees 

 with Lord Rayleigh's result, expressed in the last equation of 

 his paper on " The Stability or Instability of certain Fluid 

 Motions" (Proc. Lond. Math. Soc. Feb. 12, 1880). The 

 integral, but now with only two arbitrary constants (c , c x ), 

 is still given in ascending powers of y by (67) and (68), 

 which, with fi = 0, and the thus-simplified values ofe,f, g put 

 in place of these letters, becomes 



-i[(i + 2)0' + l)a>c i+2 + (i + l)im/3c i+1 ] 



+ [~i(i-l)mc + h~jc. + kc._ 1 + lc i _ 2 = . (72). 



For very great values of i this gives 



G)C i+2 + m@c i+1 —±mcc. = .... (73j, 



which shows that ultimately, except in the case of one 

 particular value of the ratio Cj/cq, 



<WM=?-' (74), 



where f denotes the smaller root of the equation 



G) + mj3y—fyncy 2 = (75). 



Hence there is certainly not convergence for values of y 

 exceeding the smaller root of (75), and thus the proof of 

 stability is lost. 



46. But the differential equation, simplified in (71) for the 

 case of no viscosity, may no doubt be treated more appro- 



