434 Lord Rayleigh on the Resistance of Fluids. 



Bigned to the blade is considerable and is performed rapidly, 

 the greatness of the increase in the pressure will be asto- 

 nishing until its true meaning has been realized. Utilizing 

 this proposition, many boatmen, when rowing a heavy boat 

 with narrow-bladed oars, were in the habit of alternately raising 

 and lowering the hand with a reciprocating motion, so as to 

 give an oscillatory dip to the blade during each stroke, and 

 thus obtained an equally vigorous reaction from the water with 

 a greatly reduced slip or stern ward motion of the blade." 



It is not difficult to see that in the case of obliquity we 

 have to do with the whole velocity of the current, and not 

 merely with the resolved part. Behind the lamina there 

 must be a region of dead water bounded by a surface of dis- 

 continuity, within which the pressure is the same as if there 

 were no obstacle. On the front face of the lamina there must 

 be an augmentation of pressure, vanishing at the edges and 

 increasing inwards to a maximum at the point where the 

 stream divides. At this point the pressure is ^oV 2 , corre- 

 sponding to the loss of the whole velocity of the stream. It 

 is true that the maximum pressure prevails over only an infi- 

 nitely small fraction of the area ; but the same may be said 

 even when the incidence of the stream is perpendicular. 



The exact solution of the problem in the case of two dimen- 

 sions which covers almost all the points of practical interest, 

 can be obtained by the analytical method of Helmholtz and 

 KirchhofF*. If an elongated blade be held vertically in a 

 horizontal stream, so that the angle between the plane of the 

 blade and the stream is a, the mean pressure is 



" Sina pV, (3) 



varying, when « is small, as sin a, and not as sin 2 a. The 

 proof will be found at the end of the present paper. 



The fact that the resistance to the broadways motion of 

 a lamina through still fluid can be increased enormously by 

 the superposition of an edgeways motion is of great interest. 

 For example, it will be found to be of vital importance in 

 the problem of artificial flight. 



According. to the old theory the component of resistance 

 transverse to the stream varied as sin 2 a cos «, and attained its 

 maximum for « = 55° nearly. The substitution of expression 



* Formulae (3) and (4) were given at the Glasgow Meeting of the 

 British Association. I was then only acquainted with Kirchhofi's " Vor- 

 lesungen iiber niathematische Physik," and was not aware that the case 

 of* an oblique stream had been considered by him (Crelle, Bd. 70, 1869). 

 However, Kirchhoff has not calculated the forces ; so that the forjnulse are 

 new. 



