204 Of th SOLIDS 
Let it, for inftance, be required to find a folid given in mag'= 
nitude, fuch, that from all the points in it, ftraight lines being 
drawn to any afligned number of given points, the fum of the 
{quares of all the lines fo drawn fhall be a minimwn. It will 
be found, by reafoning as in the cafe of the folid of greateft at. 
traction, that the fuperficies bounding the required folid muft 
be fuch that the fum of the {quares of the lines drawn from any 
point in it, to all the given points, muft be always of the fame 
magnitude. Now, the fum of the fquares of the lines drawn 
from any point to all the given points, may be fhewn by 
plane geometry to be equal to the fquare of the line drawn 
to the centre of gravity of thefe given points, multiplied by the 
number of points, together with a given fpace. The line, there- 
fore, drawn from any point in the required fuperficies to the 
centre of gravity of the given points, is given in magnitude, 
and therefore the fuperficies is that of a {phere, having for its 
centre the centre of gravity of the given points. 
Tue magnitude of the {phere i is next determined from the 
condition, that its folidity is given. 
In general, if x, y, and z, are three rectangular co-ordinates 
- that determine the pofition of any point of a folid given in 
magnitude, and if the value of a certain function Q: of x, y and 
z, be computed for each point of the folid, and if the fum of 
all thefe values of Q added together, be a maximum or a mini- 
mum, the folid is bounded by a fuperficies in which the func- 
tion Q is every where of the fame magnitude. \ That is, if the 
triple integral f & hs 3 {Qz be the greateft or leaft poffible, 
the fuperficies bounding the folid is fuch that Q = A, a con- 
ftant quantity. 
Tue fame holds of plane agnehe the propofition is then 
fimpler, as there are only two co-ordinates, fo that J a i Q 7 is 
the 
