496 Prof. Sylvester on the Pressure of Earth 



line of thrust and the line thrust upon stand in a reciprocal re- 

 lation to each other. 



I may add the cursory remark as regards the value of the 

 total thrusts in the case more immediately before us, that [as 

 is apparent from the equation (a)] they will be represented in 

 relative magnitude by the radius vector drawn in the direction 

 of the line thrust upon, to meet, not the ellipse of pressures just 

 described, but another ellipse whose major and minor axes are 

 to one another in the duplicate ratio of the other two. 



If we wish, however, to present the above results in a form 

 more immediately translateable into the actual case of nature, 

 I mean that of space with three dimensions, it becomes expe- 

 dient to use a different ellipse, or rather the same ellipse in 

 another position, to represent the stress at any point. 



In the equations above found, connecting N and F with 

 P, Q, R, is the angle made with a fixed axis, not by the line 

 of pressure It, but by the element on which this pressure is 

 exerted. Let <f> be the angle made by the pressure itself, so that 



<£ = 6 + ^, then we have 



N = P (cos (f>Y- 2Q sin <f> cos <j> + R (sin <£) 2 



F=(P-R)sin<£cos<£. 



And the same process as has been already employed will serve 

 to show that we may construct an ellipse such that the inverse 

 square of the radius vector in every direction may represent the 

 magnitude of the pressure in that direction (i.e. the magnitude 

 of the normal part of the thrust upon the element perpendi- 

 cular to that direction), and in this ellipse the radius vector 

 and perpendicular to the tangent at each point will represent 

 the corresponding directions of pressure and thrust, which 

 obviously will coincide for the directions of greatest and least 

 pressure. 



If, now, we go out into space of three dimensions, it will 

 readily be anticipated, and may easily be proved, that an ellip- 

 soid whose radii vectores represent the relative magnitudes of the 

 inverse square roots of the pressures takes the place of the 

 ellipse, the thrusts and pressures correspond respectively (in 

 direction) to the normal and radius vector at each point, and in 

 three directions, at right angles to each other, these latter come 

 together. 



It is desirable that the reader should bear in mind that the 

 ellipse of which I have spoken is in fact only a principal section 

 of this ellipsoid. The assumption which (following in the track 

 of my predecessors) I shall make, that the greatest energy of 



