434- Rev. H. Moseley on the Theory of Resistances in Statics. 



In a paper inserted in the Memoirs of the Academy of St. 

 Petersburgh — Novi Commentarii, torn, xviii. — entitled, " De 

 Pressione Ponderis in Planum cut incumbit" Euler has investi- 

 gated the conditions of the equilibrium of a heavy mass sup- 

 ported upon a given number of points in the same horizontal 

 plane, and upon continuous and edged bases of given geome- 

 trical forms and dimensions. As the foundation of this in- 

 vestigation, he has assumed the principle, that if the surface 

 on which the body rested were elastic, each point of support 

 would sink to a depth proportionate to the pressure it sustains. 

 Taking, then, the actual surface on which the mass rests for 

 the plane of xy 9 and assuming 



z = ax + $y + y 



to be the equation to the plane in which the points of support 

 would be found on the hypothesis of an elastic surface, z is 

 proportional to the pressure upon that point of support whose 

 coordinates are xy. And, this admitted, the constants a, /3, y, 

 may be determined by the known conditions of equilibrium. 



To this hypothesis, which is grounded on no experimental 

 fact or analogy, there is an objection, apparent on the very 

 face of it. It is this : "The forces by which the body is sup- 

 ported when it has at length attained its position of equili- 

 brium on the elastic surface, are not the same with those by 

 which it is sustained on the perfectly hard surface. Euler 

 has foreseen this objection. 



He thus speaks of it : " Neminem autem pannus ille pres- 

 sioni cedens offendat, etsi enim illi mollitiem quandam tribui- 

 mus, earn tamen quousque libuerit diminuere licebit; ita ut 

 tandem indolem soli illius, cui pondus revera insistit adipis- 

 catur." 



Now although it be admitted that the actual displacements 

 produced by the sinking of the points of support, may be di- 

 minished to any required extent by increasing the tension of 

 the surface, yet it is no less certain that the relative displace- 

 ments of these points cannot be affected by that process. So 

 that whatever error the supposition of a perfectly yielding sur- 

 face would introduce into the relation of the supporting forces, 

 the same error remains in that deduced on the hypothesis of 

 an elastic surface of extreme tension. 



There is one case of the general proposition which is in 

 this paper solved by Euler, upon known and very elementary 

 principles. It is that of a mass supported upon three points 

 in the same plane, but not in the same right line. He has 

 shown that if the point of intersection of the vertical through 

 the centre of gravity with the plane of support, be joined with 



