BY E. L. PIESSE, B.SC, LL.B. 69 



the same for all parties. This condition is expressible in 

 the form that 



(- - '-) 



shall be a minimum. This expression can be written in 

 the form 



^C-^')'-*' c^) 



37. For a, y, z, we now have the equation 



(•! _ ™)' + 0^ - "')' +(-- -)' = A'' • • . («) 



\p V ^ ^q v^ \r vy 



As before, the minimum value of k^ will be zero, and 

 then X = Xq, y = v/^, z = z^; but in this case x^ y, z will 

 not usually be integers. 



For other values of k"^ , (8) is an ellipsoid, having its 

 centre at I, and intersecting ABC in an ellipse whose 

 centre is /. As k"^ increases from 0, the ellipsoid expands 

 from 7, and is cut by ABC in a gradually increasing 

 ellipse. 



38. The solution of the problem is similar to the case of 

 the circle. As before, we plot the ideal point /; if this is 

 a node, the co-ordinates {x^^, y^^, z^) of the node are the 

 numbers of members for the parties. 



If I is not a node, we have to select the node which is 

 first touched by the gradually increasing ellipse. This is 

 not necessarily the nearest node, and to determine which it 

 is we must ascertain the direction of the longer axis of the 

 ellipse. The direction of the longer axis may be calculated ; 

 but a simpler way is to project the triangle ABC so that 

 the ellipse becomes a circle. 



39. To see the effect of this projection, change the centre 

 to the ideal point I ; the ellipsoid (8) is then 



X* v^ z^' 



— . + „"^i + — . = *^ ^ • ■ • (9) 



•^0 yo •^0 



Stretching lengths parallel to the axes in the ratios, 

 1 : Xq, 1 \ y^, \ : 2^, respectively, (9) becomes a circle, 

 and, as in § 32, the solution is given by the node nearest to 

 I . We have next to calculate the lengths of the sides a, h, c 



