390 Mr. 0. Fisher on the Thermal Conditions and 

 has for the coordinates of its vertex, 



2m 



2vr 



The lotus rectum — is independent of p and w y and may 



s 



be taken as constant. Supposing that we draw the curve with 

 assigned values of p and w, then the ordinate /3 to abscissa a 

 will give the difference in thickness between the strata at A B 

 which are derived from two contiguous rings whose widths 

 are w and w + a. 



If we put a = 0, then /3= — ; so that the height at which 



the curve cuts the axis of /3 is independent of p, except so far 

 as w depends upon p. The points at which the curve cuts the 

 axis of ol are given by the relation 



ct=p-2w±s/(p-2wy-2iv 2 . 



This must be always positive. 



Taking the smaller value, and observing that, except near 

 the pole, p — 2w is much greater than w, we have, expanding, 



...2 



nearly, 



p — 2w 



which, except near the pole, is much smaller than w. As 

 soon as a exceeds this value, /3 will become negative. 



The greatest negative value of /3 will be attained at the 

 vertex, where 



ot = p — 2w. 



In this case it will be found, by reference to the distances 

 measured from the pole, that this value of a would carry the 

 ring whose width is w + a up to the pole itself. 



We can now perceive in a general way how a decrease or 

 an increase in the width of the rings as they approach the pole 

 would affect the difference in the thickness of successive strata 

 at AB. 



The latus rectum, of the parabola being constant, the curve 



always maintains the same size. We have therefore only to 



draw it with its axis vertical, and to place the vertex in the 



position corresponding to the assumed values of p and w. It 



u? 

 will then necessarily cut the axis of /3 at the height — above 



the origin ; for this is the value of the ordinate corresponding 

 to a = 0. This shows that, if the rings were of uniform width, 

 the difference in thickness of the annual strata at A B would 



