78 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



the theory of a complex variable, the problems treated are those of 

 two dimensional flows, and here the whole fluid is divided by a sur- 

 face of discontinuity (for velocity) into a moving and a stationary 

 portion with the stream lines in the moving portion diverging instead 

 of closing in behind the obstructing object. This state of affairs may 

 be found in nature to a certain approximation in the case of jets. It 

 is clear that the theory of the continuous solution is entirely inapplic- 

 able in the discussion of the pressure exerted on the disk by the fluid, 

 and that for that problem the discontinuous solution would be neces- 

 sary. And it is equally clear that for a discussion of the possibility 

 of a critical velocity the discontinuous solution is disadvantageous, 

 and the continuous solution must in the present state of our knowl- 

 edge be used for what information it may afford. 



We shall henceforth assume that the fluid is incompressible, that 

 the motion is irrotational, symmetric fore and aft of the disk, and 

 identical in all planes through the axis of the disk ; that the eddies 

 which are known to form are to be disregarded ; and that the viscosity 

 may be neglected. We know that the ordinary continuous symmetric 

 solution must break down at least as soon as cavitation appears, and 

 we may with reasonable safety assume that the velocity u=U oi the 

 stream sufficient to induce incipient cavitation at the edges of the 

 disk will be an upper limit for the critical velocity found by Mr. 

 Hunsaker. (There is unfortunately no assurance that the upper 

 limit may not seriously exceed that critical velocity.) Now if the 

 disk has a perfectly sharp edge, cavitation will take place for any 

 velocity of the general stream, no matter how small that velocity 

 may be. To get any result of value we must therefore replace the 

 disk by a body of rounded contour, such as an ellipsoid of revolution. 

 The theoretical problem which we shall solve is therefore this : To 

 find the velocity U at which cavitation begins in the case of an 

 extremely flat ellipsoid of revolution whose axis coincides with the 

 general course of the stream; and for this we shall determine a very 

 simple approximate expression in terms of the axes of the ellipsoid. 

 For an ellipsoid 6 inches by 1/16 of an inch we shall find U = 22 

 foot-seconds, or thereabouts, and this result will be discussed in the 

 light of the experiments. 



As the motion is irrotational, by assumption, there is a velocity 

 potential cf>, and as the density is assumed constant, the velocity 

 potential satisfies Laplace's equation, 



dx- d\- dz^ 



