494 Prof. Sylvester on the Pressure of Earth 



from it by an infinitely small quantity ; in like manner /' may 

 be taken as the face-force on Q R, PS respectively. Hence the 

 couples whose moments are (f. Q R) x Q P and (/' . P Q) QR re- 

 spectively must be equal and opposite, or in other words, / being 

 understood to act to or from Q, according as f acts to or from Q, 

 we must have /=/'. To fix the ideas, conceive the face-forces to 

 tend towards Q. Let us now consider the equilibrium of the 

 triangular molecule P Q R. Call the pressure on P Q (R), the 

 pressure on RQ(P) the face-force on PQ or QR (Q). In 

 comparison with the thrusts on the faces of our triangular mole- 

 cule, gravity or other impressed forces may be neglected as giving 

 rise to quantities of an inferior order of smallness. 



Let QP, QR be regarded as two fixed rectangular axes, and 

 let QPR=0. Let the pressure and face-force on PR (always 

 understanding thereby the units of such forces) be called N and 

 F respectively (F, to fix the ideas, being taken to act from P to 

 R), Then resolving the forces perpendicular to PR, we obtain 



N . PR=R . PQ . cos QPR+ P . QR cos QRP 



+ Q.PQ sinQPR + Q.QRsinQRP, 

 or 



N = R (cos 6f + 2Q cos 6 sin 6 + P (sin Of ; 



and resolving parallel to PR, we have 



F x PR = R . PQsin QPR-P . QR sin QRP 



+ Q . PQ cos QPR - Q . QR cos QRP, 

 or 



F = (R-P)sin0cos0. 



Imagine now QRR to be represented by a single point 0. 

 R, P are respectively the pressures, and N the face-force (" units 

 of pressures and of face-force ") on elements drawn in the ortho- 

 gonal directions OX, Y ; N the pressure, and F the face-force 

 on an element drawn in the direction OP, making an angle 6 

 with OX. Obviously, therefore, if we draw in all directions 

 from lines whose lengths are as the inverse square roots of 

 the pressure-part of the thrust acting on those lines, calling 

 the length of line corresponding to 0, r, we have 



-\ = R (cos 0) 2 + 2Q sin 6 cos 6 + P (sin 0) 2 , 



R, Q, P being constant quantities. 



Consequently the locus of the extremities of these lines is a 

 conic ; and taking new axes of coordinates in the directions of 

 the principal axes of this conic, and understanding by R and P 

 the pressures perpendicular to those axes respectively, the equa- 

 tions obtained assume the form 



