MH.II>>. ON SURFACt>. 367 



gravity be at the balancing point, above it. or below it. This 

 is evident from the equation of the curve, 



where it i- manifest thai j maybe taken arbitrarily, positive, 

 Q< gative or nought, without producing any other change than an 

 elevation or depression of the origin, while the differeni values 

 and signs which we may ascribe to a and 8 will furnish us 

 with areolas of every variety of curvature. This advantage is 

 however unimportant in the present inquiry, which is rather to 

 n.iin the results of the general problem than to enter into 

 a detailed examination of each particular case. Resuming 

 therefore the expressions (30) before obtained for ellipsoids <>n 

 ellipsoidal surfaces, and observing that the quantities k and /■• 

 in the case of small oscillations become constant and equal to 

 the fixed and moveable vertical semi-axes, retaining at the same 

 time the symbols e", x". z,. £ . £ . in order to permit without 

 further substitutions the application of the usual formulas, 

 the second triplet of equations (30) furnish, when the areola of 

 support is spherical, whatever he the form of the areola around 

 the balancing point of the oscillating body, 



/,• . / ./• = /,-. / ( ax — <i -r ■+■ '/ '• ) . 



h liy = hJlhx + Vx -+■!> .r ). 



I:('z = A-. /(<•./• + cj' -+- c ./• ). 



Bj means of equations (3) these become 



wy,A,X = j.+f . /• — It . 



vy.B.y = y.-hv, , or y = rm . 



C,z, = *, -h £ ; r, = n{ 



Substituting the values of./-, y. z in equations (31) and em- 

 ploying the following abridgments, 



\ hi. hi. — 5 \ 



