is 



Scientific Proceedings, Royal, Dnhlin Society. 



= xP - \Pdx ; but, oi 0, X = 0, and, oi A, P = ; therefore, xP 

 vanishes at each limit, and /. , as P =^ gp%^, %i^Xy = - Iz^dx ; but dA, 

 the area of UW, any element of the parabola contained between 

 the curve, the element dx, and the lines through its extremities 

 parallel to s, is equal to - s sin (^tdx, while ~, the ordinate of the 



centre of gravity of this element, = ^ ; therefore 



ZiXi sin = - sin ^f) JsViC = ^zdA = '2 IzdA = 2Azi, 



or 



2AI 



Xi = 



sm^ 



As a verification of this result, let h vanish ; the hyperbola then 

 ■reduces to its asymptotes ; the branch A V becomes the straight 



T, 



Fis-. 7. 



line through A, making, with the vertical, the angle 7 - | (this 

 also appears geometrically ; for, if (fig. 7) 



but 



= ^+-,anda = ^-^y, 



2A = Hi Si sin (p, and Si = ^ 



o 



