364 ON THE MOTION OF 



p, q and r are best obtained by means of formulas (6). They 

 furnish immediately (32) 



p = ra" + db", q = rb" — da". 



The same equations give da = bdR, db = — adR, which, 

 integratedin conjunction with a* -\- V = i, give us a = cosR, 

 h = — sin R, the angle R being counted from the axis of x. 

 The nine cosines then become 



a =cosi?, b = — sinK, c = b'sinR — a'cosR. 

 a' = sh\R, b' = cosR, c 1 = — b'cosR — «"sin/?. 



a"=a"; b" = b"; c" = i. 



From equations (S) and the equations of the surfaces we ob- 

 tain 



X = a , £ = a — y ( , 



_ yi II 



z, = y ; £ = a — y, . 



The analysis gives these constants the double sign, which I 

 omit, as in case of application it will always be immediately 

 obvious which will be affected with -+- and which with — . 

 Thus if both areolas are concave upward, and the eentre of 

 gravity of the oscillating body is above the point of contact 

 and below the centre of the figure which osculates with the 

 areola of support, then the signs remain as above, the ellipsoid 

 or elliptical paraboloid being in such a case the proper oscula- 

 ting figure. If, as in the common pendulum, the point O, is 

 below B,, and the two areolas are still concave upward, the 

 osculatrix of the areola at B, must be an hyperboloid or el- 

 liptical paraboloid with the point O, taken in the prolongation 

 of the axis, and the constant a" would change its sign. If 

 the pendulum were hung upon a fixed cmmdus interlinking 

 with another annulus at the upper extremity of the pendulum, 

 both areolas would then become discurvate and the osculating 

 figures would be either single-napped hyperboloids or hyper- 



