138-139] INTRODUCTION OP FLUX-COORDINATES. 213 



We now recur to the formula (4) of Art. 132. The variations 

 Af, A?;, A f being subject to the condition of incompressibility, the 

 part of the sum 



2m(fAf + ?}A?7 + A) (3) 



which is due to the fluid is, in the present notation, 



J J J \ cLx (by (Hz / 



n&)dS 



where the surface-integral extends over the bounding surfaces 

 of the fluid, and the symbols K, K', ... denote as usual the cyclic 

 constants of the actual motion. We will now suppose that 

 the varied motion of the solids is subject to the condition that 

 Ag l5 Ag 2 , ... 0, at both limits (t and ^), that the varied motion 

 of the fluid is irrotational and consistent with the motion of the 

 solids, and that the (varied) circulations are adjusted so as to make 

 A^, AX', . . . also vanish at the limits. Under these circumstances 

 the right-hand side of the formula cited is zero, and we have 



(5). 



If we assume that the extraneous forces do on the whole no 

 work when the boundary of the fluid is at rest, whatever relative 

 displacements be given to the parts of the fluid, the generalized 

 components of force corresponding to the coordinates %, %',... will 

 be zero, and the formula may be written 



h...}cft = (6), 



where 



dT A dT A dT A dT A dT A d? 7 



Al =-T-T Afl^H pr Ao'a-f-. .. + ^ Atfj-f- -= , A^-f. .. + 7 Ay-| r/ Av 



a^! a^ c^^i ^^2 ^% c^% 



+ (7). 



139. If we now follow out the process indicated at the end of 

 Art. 133, we arrive at the equations of motion for the present 

 case, in the forms 



J jrn jm -j -jrn jm 



CL d-L tt-L .- d CLJL a,2 ~ 



dtdfr dq ' dtdq 2 dq. 2 2 ' 





L _ 

 dtdx dtdtf 



