Of GREATEST ATTRACTION. 205 



the quantity that is to be a maximum or a minimum, and the 

 line bounding the figure is defined by the equation Qj= A. 



All the queftions, therefore, which come under this defcrip- 

 tion, though they belong to an order of problems, which re- 

 quires in general the application of one of the moft refined in- 

 ventions of the New Geometry, the Calculus Variationum, form 

 a particular divifion admitting of refolution by much Ampler 

 means, and directly reducible to the conftruction of loci. 



In thefe problems alfo, the fynthetical demonflration will be 

 found extremely fimple. In the inftance of the folid of great- 

 eft attraction this holds remarkably. Thus, it is obvious, that 

 (Fig. 1.) any particle of matter placed without the curve 

 ACBH, will attract the particle at A in the direction AB, lefs 

 than any of the particles in that curve, and that any particle of 

 matter within the curve, will attract the particle at A more 

 than any particle in the curve, and more, a \ fortiori, than any 

 particle without the curve. The fame is true of the whole 

 fuperficies of the folid. Now, if the figure of the folid be 

 any how changed, while its quantity of matter remains the 

 fame, as much matter muft be expelled from within the fur- 

 face, at fome one place C, as is accumulated without the fur- 

 face at fome other point H. But the action of any quantity of 

 matter within the fuperficies ACBH on A, is greater than the 

 action of the fame without the fuperficies ACBH. The folid 

 ACBH, therefore, by any change of its figure, muft lofe more 

 attraction than it gains ; that is, its attraction is diminifhed by 

 every fuch change, and therefore it is itfelf the folid of great- 

 eft attraction. Q^ E. D. 



Cc 2 XIL. 



