Prof. Challis on the Principhs of Hydrodynamics. 449 



cates a general law of the spontaneous mutual action of the parts 

 of the fluid. I have shown in the Philosophical Magazine for 

 May 1849 (p. 363), that the same result is obtained by supposing 

 the arbitrary functions in the general integral to be arbitrary 

 constants. It may be worth while to indicate still another pro- 

 cess by which the equation (p) may be deduced. Let 



^ + 2/^ — l=At, x-y\^^^ = v; 

 and m order to get rid of the impossible quantities, make F and 

 G the same functions in the general integral of (v). Then by 

 supposing F(/^)=Ae^^ and G(v)=Ae'^", e being the base of the 

 JNapierian system of logarithms, the following exact value of/ is 

 found: C -^\ 



/=2Ae^'"^>'cos(A+|)y. 



LetA' = rcos^, y = r^m6; k~~^m, ^+ j = «; and expand 

 the right-hand side of the above equation, viz. 



-cAe COS nr sm v, 



in terms arranged according to the dimensions of m and n. Then 

 tf, for the reason already alleged, the summation S .fh6 be taken 

 from ^ = to ^=27r, and the constant A be determined so that 

 the sum shall be unity when ?-=0, the result is 



which, since m^-n^= -4e, is independent of the arbitrary quan- 

 tity k, and is plainly identical with the right-hand side of (p). 



The values that have now been obtained for ^ and / define 

 precisely the motion along, and perpendicular to, the axis of 

 rectilinear motion. 



It may here be remarked, that a discussion of the equation (p) 

 shows that / has an unlimited number of maximum values which 

 become less as the distance from the axis is greater, and finally 

 vanish at an infinite distance. Hence at an infinite distance f 

 df ,df •" 



dx' dij ^^"'^"^ ^"" '^y consequence the velocity vanishes. 



Hence also the condensation vanishes. Thus the supposition 

 already made, that the arbitrary quantity F(0 is equal to zero, 

 18 shown to be legitimate by the result of the preceding inves- 

 tigation, no part of which depends on that supposition. The 

 condensation (cr) is therefore correctly given by the equation («), 

 from which it is readily seen that we have also 



d:^a . <^V , 



^ + ;^ -t-4..r = (,) 



As the equation (tt) is a unique and exact integral of (v) 

 Phil. May. S. 4. Vol. 4. No. 27. Bee. 1852. 2 G 



