J"= 



Lord Rayleigh on the Resistance of Fluids. 439 



-jr. Thus, since J is real, -k£=p= -j- ; and therefore the in- 

 tegral may be replaced by 



'*!(<:- J)** ..... (c) 



in which all the elements are to be taken positive. 

 From the form of £ in (B), it appears that 



l(H) = \/( cosa+ ^) 2 - L - • < D) 



The width of the lamina I is J £dco, where the limits of integra- 

 tion are such as make 



1 



cos«H = ±1. 



sj CO 



The integration may be effected by the introduction of a new 

 variable /3, where 



j3 = sin 2 a. s/ co — cos a, 



and the limits for ft are ±1. Thus 



f( 



1 \ . /3 2 cosa , 2/3 



cos u ^ — ) aco = -^—. 1- -i—7 h const. : 



s/co) sin « sin* a 



and therefore between the limits ± 1 we have 



4-f- sin 4 a. 



The second part of ? may be written \/l— /3 2 -r- sin u^'co, 

 giving the integral 



i: 



day ., m /3s/l-f3*sm-i/3 L 



v 1 — p- = — ^7^— h _... 3 + const. 



sm a. ^/co sir a sm° a 



Thus the complete value of z between the limits, or l, is 



4 7T 4 + 7rsina ^ 



t=-r-T h -7—5 — = r-r — . . . . (Hi) 



sm a sir a sin* a v y 



By (C) and (D) the whole pressure on the lamina is repre- 

 sented by the second part of I in (E), or tt-t- sin 3 a ; so that 

 the mean pressure is 



7r 4 + 7r sin a ir sin a 



4 + 7r sin a 



as was to be proved. 



Again, the elementary moment of pressure about z = is 



l/„ 1\_ 2v/l-/3 2 _- 



± o ( K— -p ) rf« • z— — . , z dp. 

 2 v V sm * 



