Remarks on the " Principle of Least Pressure" 1 95 



in the practice of statics ; and one which, although it has never 

 yet received a complete and satisfactory discussion, yet lies 

 on the very threshold of the science of mechanics. 



The demonstration which I have given of the principle of 

 least pressure is shortly this. ; 



" Let there be conceived a system of forces ot which a 

 certain number are given in magnitude and direction, and the 

 rest are supplied by the resistances of as many fixed points. 

 Also let the points of application of the forces of the system 

 be supposed to be given. 



« Let A designate the given forces of the system, B the re- 

 sistances, and C any other system of forces which, being ap- 

 plied to the same points with the forces of the system B, would 

 maintain the equilibrium. Also let the system C be supposed 

 to replace the system B. - m 



" Now each force of the system C, under these circumstances, 

 just sustains and is equivalent to the pressure propagated to 

 its point of application by the forces of the system A ; or it is 

 equivalent to that pressure, together with the pressure pro- 

 pagated to its point of application by the other forces of the 

 system C. . ■ ,. 



" In the former case it is identical with the corresponding 

 resistance of the system B. In the latter case it is greater 



than it. «,'.-• 



" The sum of the forces of the system B, each being consi- 

 dered a function of the coordinates of its point of application, is, 

 therefore, a minimum." _ 



Mr. Earnshaw's first remark upon this is, that, speaking 

 generally, it is impossible that each force of the system C 

 shall just sustain only such pressures as are propagated to it 

 by the system A ; unless either that the system C consists ot 

 but one such, or that all the forces of which it consists are 

 parallel. For if there be more forces than one in the system 

 C, and if they are not parallel, their actions ramt of necessity 

 mutually propagate pressures. Wherefore," &c. 



Now to this objection I have the following answer to 

 make. Each force of C sustaining as well the pressure pro- 

 p :l .rated from the system A, as that arising out of the mutual 

 action of the forces of its own system, let it be resolved into 

 two others, one equal and opposite to the former pressure, and 

 the other to the latter. Call the system of forces thus 

 equal and opposite to the pressures propagated from A, and 



composing which, there do not enter two or more resistances. The general 

 solution of that case cannot be effected by the known cond.t.ons ol equili- 

 brium. C2 



