202 SEARLE. 



VIII. 



Area included by the curve and its axes. 



The value of U, as above, is VKL""- cZ"-) V - {Z + L cos voY]. 

 The integration of UdZ between the limits Z = K and Z = should 

 give twice the area enclosed by the curve and its axes, wath the addi- 

 tion of the rectangle U sin vq cos vq. When w = 1 or tv = 2, either 

 sin fo or cos ^o has the value 0, and this rectangle disappears. The 

 integration, in other cases, if not directly practicable may be effected 

 by the customary indirect methods. It is not here discussed. 



When w = 1, ir^= [(A + C) - CZ/AY- [Z - {A + C)]\ that is 

 (A-\-Cy-2{A+C)CZ/A+C^Z'/A'-Z'-^2{A+C)Z- {A-\-C)\ and 

 after reduction [(C^- A^)Z'— 2A(A -{- C) (C - A)Z\I A^= {2AB'Z- 

 B^Z^)/A^ 



In the circle with radius A, the squared perpendicular which corre- 

 sponds to U is expressed by A"^- {A - Zy= 2AZ - Z^; so that V^ 

 has the ratio B^/A^ to this squared perpendicular, and U has the ratio 

 B/A to the perpendicular itself, whence we find the elliptic area in the 

 usual way, when apastron occurs. 



When w = 2, U^=A^-(A''-B^)ZyB^-\-Z^=[A''B''-A^Z^-\-B^Z'~- 

 B^Z^]/B\ which becomes A^B"- - Z") / B\ In the circle with radius B, 

 the squared perpendicular which corresponds to ?7 is B"— Z", so that 

 U has the ratio A/B to that perpendicular, and the elliptic area is 

 found as before. 



IX. 



Value of the radius vector at which the forces are equal. 



It is now desired to find the value of the radius vector P. When 

 R = P, the variation of R v>ith. respect to time is at a maximum. 

 Denoting time by t, we now require a maximum value for dR/2dt, 

 which may be expressed either as dR/U dZ or as dR/RHv. In the 

 first case, the independent variable will be Z; in the second, cos v 

 may be more convenient than v itself. 



A maximum of dR^/UHZ~ will occur with that of dR/UdZ. In the 

 differentiation required to determine this maximum, it must be 

 noticed that U^ increases while Z diminishes, and hence that d{U^)/dZ 

 is negative. 



