IV. 



ON A THEOREM RELATIVE TO MOVEMENTS THAT ARE GEOMETRICALLY 

 SIMILAR IN FLUID BODIES, TOGETHER WITH AN APPLICATION TO 

 THE PROBLEM OF STEERING BALLOONS.* 



By Prof. H. vox Helmholtz. 



The laws of motion of cohesive and non-cohesive fluids [namely, liquids 

 and gases] are sufficiently well known in the form of differential equa- 

 tions, that take into consideration not only the influence of exterior forces 

 acting from a distance, as well as the influence of the pressure of the 

 fluid, but also the influence of the friction [namely, both internal and 

 external frictions, or both viscosity and resistance]. When in the 

 application of these equations one remembers that under certain cir- 

 cumstances [namely, wherever a continuous motion would give a nega- 

 tive pressure] there must form surfaces of separation with discontinuous 

 motion on the two sides, as I have sought to prove in a previous com- 

 munication to this academy,t then will disappear the contradictions 

 that by neglect of this consideration have hitherto been made to appear 

 to exist between many apparent consequences of the hydro-dynamic 

 equations on the one hand and the observed reality on the other. In 

 fact, so fiir as I see, there is at present no ground for considering the 

 hydro-dynamic equations as not being the exact expression of the laws 

 controlling the motions of fluids. 



Unfortunately it is only for relatively few and specially simple ex- 

 perimental cases that we are able to deduce from these differential 

 equations the corresponding integrals appropriate to the conditions of 

 the given special cases, especially if the nature of the problem is such that 

 the internal friction [viscosity] and the formation of surfaces of discon- 

 tinuity can not be neglected. The discontinuous surfaces are extremely 

 variable, since they possess a sort of unstable equlibrium, and with every 

 disturbance in the whirl they strive to unroll themselves; this circum- 

 stance makes their theoretical treatment very difficult. Thus it happens 



* From the Monatshericlite of the Royal Academy of Berlin, June 26, 1873, pp. 501 

 to 514. WissenschaftJiche Aihandlungen, vol. ii, pp. 158-171, Berlin, 1882. 



t Berlin Monatsberichte, April 23, 1868. See also No. Ill of this collection of Trans- 

 lations. 



67 



