Brard 



This possibility comes from the property of the vector W(M), having its 

 origin in M, and defined by the formula 



W(M) = 



I 





dD.(iJ.) , 



where V(/x), which has its origin in ^, is continuous with respect to the point ^ 

 which describes the volume fi; its first derivatives also are assumed to be con- 

 tinuous; moreover 



div V(m) = . 



Taking for Q the space n^ exterior to the body, and for V(/l(.) the absolute 

 velocity V^ = - grad <l)o of the fluid, then using the equation 



curl curl W = grad div W - VW , (1) 



where 



we obtain 



Bx2 3y2 Bz2 



477 



curl 



nAVo(/x) 



^M dS(M)f-4^grad 



iM 



I Vq(M), when M is in 

 dS(/x) = ^ fig , 



I 0, when M is in fi^ , 



fi^ being the volume inside the body. 



Taking now for n the volume n^ , and for \(i-l) the absolute velocity of m 

 considered as at rest with respect to the body, we have VCm) = V^C/x) with 



\^(fx) = U+QAO/x, curlVE=2Q, 



and 



477 



curl ■{ - 



hAVeCm) 



dS(M) 



. l\\ curl V, 





+ — grad 



477 ^ 



I 



hVeCm) 

 AiM 



dS(/x) 



when M is in 0.^ , 

 e ' 



Vp when M is in n. 



By addition, we find, with V^ = V^ - V^ = the relative velocity of the fluid, 



822 



