of the Motion of Incompressible Fluids. 127 



for it has been long ago proved that every impossible quantity 

 may be put under the form a ± b \/ — I. 



g= i/-lF(^+3/ V'-I) ->/-!/' (^-3/ ^-1) 

 = A ^~i _ B - A' v'~l - B'. 



Hence ^ and — cannot both be possible unless B = B', 



and A = A', that is, unless F' andy be the same functions. 

 As the direction of the axes is quite arbitrary, suppose ^ = 0. 



Then ^ = 2 F (x), and ^ = 0. This proves that the velo- 

 city is directed to or from the origin of coordinates, and is 

 equal to twice a function of the distance of the same form as 

 P. Hence, 



Let x-i-7/\/—l=m, X — 7/ ^—1 =z )i; so that 

 2_y = (w — w) -/ — 1, 7^ = m n. 



... F {m) - F {n) = ^ . F(^^ 



As this equation is identical, F (m) is the same as s/ — x 



' ^ Q m 



YH\/mn): hence F'(v'»Jw) must = —7=. This form of 



the function evidently satisfies the equation. Hence g' = 2 F' (r) 



2 c 1 

 = — = — X an arbitrary function of ^, the same result as 



before. 



When the motion is in space of one dimension, 



Hence 1^ =/' {t) ^ + F {t) = ""-^ + F (/). 

 Substituting in the general equation (1), 



__V____-^-_l ^i). 



I proceed to a])ply this equation to a problem of consider- 

 able interest. Sujipose a vessel A BCD, roiined by (lie revo- 

 lution of a line A EH, which nmy be cithiT continuous or 

 broken, about an axis O N, to contain lluiil always retained 



at 



