204- Of the 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 

 fquares of all the lines fo drawn fhall be a minimum. It will 

 be found, by reafoning as in the cafe of the folid of greatefl at- 

 traction, that the fuperficies bounding the required folid muft 

 be fuch that the fum of the fquares 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 fliewn 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 fphere, having for its 

 centre the centre of gravity of the given points. 



The magnitude of the fphere 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 J xj yj Qjs be the greateft or leaft poffible, 



the fuperficies bounding the folid is fuch that Qjz: A, a con- 

 ftant quantity. 



The fame holds of plane figures j the proposition is then 



fimpler, as there are only two co-ordinates, fo that J x j Qy is 



the 



