260 EXAMPLES OF MAXIMA AND MINIMA, 



Multiply the first of these equations by x, the second by 



2v 



^ , and the third by z, and add ; then 



therefore X = 4 . 



* 



TT 2 2 12C * I, .3 C * 4 3C * 



Hence V = , & ^ = ^> I?' 



5. To find the maximum and minimum value of r 2 when 



r 2 =(*-a) t + ( 2 /-/3) 2 +(3-7) 1 , 

 the variables and constants being connected by the equations 



=P .- (2), 



= p (3), 



^=A=^ (4). 



[The student who is acquainted with Geometry of Three 

 Dimensions will see that (1) is the equation to an ellipsoid, 

 and (2) is the equation to a plane ; a, /3, 7 are the co-ordinates 

 of the centre of the curve of intersection of the plane and the 

 ellipsoid, and r is the radius vector drawn from the centre 

 of this curve to any point of the curve.] 



Since j- 2 is to be a maximum or minimum, we have 

 also from (1) and (2) 



at Tim t* /)/ 



(6), 



(7). 



