606 , Prof. W. M. Hicks on the 



Now X is a function of k alone. The curve is then traced 

 by giving a series of values to k, and calculating x from this 

 equation. The corresponding ordinate is found from the 



circle £• = const, or (x + a)*+y 2 = —jj- . If a complete set of 



these circles (corresponding to bipolar co-ordinates) be 

 drawn the points corresponding to given x can be found 

 graphically, although the method is not susceptible of great 

 exactness when the points are in the neighbourhood of the 

 equatorial plane. If the boundary be drawn for a given 

 value of A, that for any other suitable value can be drawn at 

 once in the following way. Suppose the new value of A is 

 /'.A. If the boundary for A cuts one of the circles in P> 

 draw PN perpendicular to the axis and on it take P'N=/'.PN. 

 The perpendicular from P' to the axis of x (or equatorial 

 plane) will cut the same circle at Q, which is a corresponding- 

 point on the new boundary. It will therefore be sufficient 

 to draw the curve accurately for a particular value of A only. 

 The curve I in fig. 2 is drawn for A = 2. A=l corresponds 

 to 6/a = *002 and is too small to give a singly-connected space. 

 A = 2 corresponds to 6/a = *368, which is far too large for our 

 approximate formulae to hold. But this is immaterial for a 

 standard curve from which to draw others for which the 

 approximation holds. For the limiting case A= 1*1741 or 

 y=*587. In general 



1 /2\ 3 - 



bja 



-L 



These give bja when / is given, or / when bja is given. 

 The table * at the end gives the values of X, log X, X 23 for 

 a series of values of k used in the calculations of this paper. 



Curves II, III, V, give the singly-connected boundaries 

 for the three cases of b/a= m l, "05, *0ll6 — the last being the 

 limiting case. They were drawn graphically from I by the 

 method indicated above. 



The areas and volumes found graphically from the curves 



are : 



Area. Vol. 



II P220a 2 2'18a 3 



III '478a 2 l-685a 3 



Y ... -268 a 2 1-362 a 3 . 



* The values of F, E are taken from the values given in Bertrand's 

 Calcul Integral. 



