78] RESISTANCE OF A LAMINA. Ill 



To find the centre of pressure we take moments about the origin. Thus 



J^x(l-lP)*dfi 



the remaining terms under the integral sign being odd functions of and 

 therefore contributing nothing to the final result. The value of the last 

 integral is |TT, so that the moment 



TT 3 cos a 



sin 3 a 4 sin 4 a ' 



The first factor represents the total pressure ; the abscissa x of the centre of 

 pressure is therefore given by the second, or in terms of the breadth, 



cos a 



This shews that the point in question is on the up-stream side of the 

 centre. As a decreases from ^n to 0, x increases from to ^l. Hence if 

 the lamina be free to turn about an axis in its plane coincident with the 

 medial line, or parallel to this line at a distance of not more than ^ of the 

 breadth, the stable position will be that in which it is broadside on to the 

 stream. 



In the following table, derived from Lord Rayleigh's paper, the column I 

 gives the excess of pressure on the anterior face, in terms of its value when 

 a = 0; whilst columns II and III give respectively the distances of the centre 

 of pressure, and of the point where the stream divides, from the middle point 

 of the lamina, expressed as fractions of the total breadth. 



The results contained in column I are in good agreement with some 

 experiments by Vince (Phil. Trans. 1798). 



