O^O TRANS. ST. LOUIS ACAD. SCIENCE. 



centric circles. If the value of ^ — , as deduced from (o), be 



Sin 6 ^^'^ 



substituted in (8), it becomes 



This is the equation of a parabola the position of whose vertex 

 is given by the conditions 



^ = 0, 



X = R', 



z = a'= *R', 



the distance z being, of course, laid oft' at right angles to the plane 

 of X, y. Revolving this parabola about its transverse axis, which 

 is parallel to the axes O and S, the paraboloid of revolution ob- 

 tained will represent the relation between a and R for every point 

 in the field. The changes which this surface undergoes during 

 an oscillation of the pendulum are very curious and interesting, 

 but it is unnecessary to enlarge upon them here further than to 

 remark that the focus of the paraboloid is always in the axis x^ 

 and its vertex is always in one of two right lines lying in a hori- 

 zontal plane, and making an angle of 30 degs. with the axis x^ 

 and intersecting at S. 



These considerations are wholly independent of the maximum 

 amplitude of swing, and also of the geometry of the pendulum^ 

 excepting so far as it is involved in the distance /. 



The concentric circles, which represent the isodynamic lines 

 of the disc pendulum, are, of course, the right sections of coaxial 

 cylinders, which represent the isodynamic surfaces of any com- 

 pound pendulum. 



When 61 = 0, these consecutive surfaces become a series of 

 vertical and equidistant planes, as is shown by equation (6). 



