Mb HOLDITCH, ON ROLLING CURVES. 69 



and an hour is sufficient to make a table for any assumed apsidal dis- 

 tances. It will be seen that if k be positive, as k increases, the curves 

 bulge at the greater apse ; if k be negative and increases, the curves 

 bulge more and more at the lower apse ; this will afterwards appear 

 from the consideration of the radius of curvature. 



Fig. (25) is an example of a two-lobed curve. 



In some cases, as in figures 8, 9, 10, the semilobe commences at the 

 minor apse by a retrograde motion of the radius vector, and terminates 

 in such cases by a retrograde motion at the major apse: for let A be 

 the value of near the smaller apse when r = b + as, and B the value of 

 ir — 6 near the major apse when r = a — as, as being a very small quan- 

 tity, then we get from the equation to the curve ; 



and therefore if k be positive, A and B are positive ; and if k be nega- 



A a 



tive, A and B will be both positive, or both negative ; for ^ = t » so that 



if a portion of the upper semilobe is below the axis at one apse, there 

 will also be a portion below at the other apse. 



As k increases the curves run into hooks, the points of which have 



a tangent passing through the centre, and there can only be two in 



/ a + b\ 2 

 each semilobe determined from the equation k t + k. (r — J = 0, for 



d9 

 at these tangent points — = 0, and this equation has only two roots, the 



sum of which is a + b and therefore in rolling they come into contact. 



If the value of k t from equation (2) be substituted in this last, the 

 distances of the tangential points from the centre are 



*±* ± \/ _ y^> + E±3 . (j-a - Vby. (5). 



