Remarks on the " Principle of Least Pressure" 203 



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Q = iP. 



This is the case spoken of before, in which the directions of 

 Q and Q' are parallel. 



Now the above results may be arrived at by another and an 

 entirely independent process of reasoning. 



Let P' and P'' each equal one half of P, and let them be 

 applied immediately above the points Q and Q'; they may 

 then be made to replace P without in the least altering the 

 circumstances of the equilibrium. Now if the direction of 

 P'Q be "within the limits of the resistance of the surfaces at 

 Q, the pressure P' will be wholly sustained by that resistance, 

 and the direction of the force Q will be in the same straight 

 line with P'Q ; the wedge sustaining no pressure whatever 

 laterally or in a direction perpendicular to PA. But if the 

 direction of P'Q be without the limits of resistance at Q, then 

 some other force must be supplied at Q, in order to maintain 

 the equilibrium. That force can only result from the action 

 of the force P" at Q'. It acts, therefore, in the line Q' Q, and 

 therefore in a direction perpendicular to PA. Also, this 

 force, resulting from the tendency of the wedge to motion on 

 the point Q", is only just equal to that tendency, or in other 

 words, it is equal to the least force which would keep that 

 point at rest. Since, then, it is equal to the least force which 

 would keep the point Q' at rest, it is also equal to the least 

 force which would keep the point Q at rest: now the least 

 force which would keep Q at rest is manifestly that which will 

 bring the direction of the resistance at Q just within the li- 

 miting angle of resistance at that point. Thus, then, it appears 

 that the directions of Q and Q' are inclined to the normals at 

 those points at angles each of them equal to the limiting angle 

 of resistance. This is precisely the result which is given us 

 at once, by the principle of least pressure. 



There are numerous applications of the principle of least 

 pressure in the theory of statics which admit of a verification 

 similar to the above. Take, for instance, the theory of the 

 arch. The pressures upon the surfaces of the abutment and 

 keystone should, by that principle, be each a minimum, subject 

 to the condition that they should be sufficient to sustain the 

 semiarch, if it formed one continuous solid, and that the pres- 

 sure on the key should be horizontal. Now the weight of the 

 semiarch being given, as the pressure upon the key dimi- 

 nishes, that upon the abutment also diminishes. Also, the pres- 

 sure upon the key tending to support either semiarch results 

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