View of the XJndulatory Theory of Light, 21 



which we will write for brevity = s^, then from these equa- 

 tions, by an easy process, on eliminating the angles, we can 

 deduce an equation of the third degree with respect to 5% 



(L-5^)(M-5^)(N-s^)-P^(L-s^)-Q^(M— 5^)~R^(N-5^) 

 + 2PQR = 0. 



It follows that each of the three real roots of this equation 

 has corresponding to it a distinct set of values of the cosines 

 of a, /3, y, and consequently a distinct straight line passing 

 through the centre determined as above : it is also shown that 

 this equation has always three real roots. 



Also, (supposing the three roots unequal^) it is proved that 

 the three lines will be at right angles to each other, and will 

 coincide in position with the three geometrical axes of the sur- 

 face represented by the equation at first assumed. 



If two, or all three, roots are equal, the author considers 

 the corresponding cases, in which two or all three of the lines 

 passing through the origin are indeterminate in position. But 

 in the former case the two, though arbitrarily placed in other 

 respects, yet lie in one plane, to which the third is perpendi- 

 cular : in the second case, they are any three lines arbitrarily 

 drawn through the origin ; and we are consequently at liberty 

 to assume them so as to be perpendicular to each other. 



When the surface is an ellipsoid the three values of s^ are 

 precisely the squares of the three semiaxes. 



Equations of Motion of a System of Molecules, 



Let us conceive a system of material molecules arbitrarily 

 distributed in space, and subject to be put in motion by the 

 force of their mutual attractions or repulsions. We will call 

 the mass of one of these molecules m, and those of the others 

 ?w, m\ ?w", &c. 



Let us first suppose the system in a state of equilibrium. 

 Referring to three coordinate axes, let us suppose the coor- 

 dinates of m to be x y z\ 



of /w, x-\- L,x y+ Ay 2+ As; 



the distance of m from m to be r : whose projections on the 

 three coordinate planes are Ax, Ay, Az; whence, on the 

 principles of solid geometry, we have 



r^ = A^-+Ay+As^ (L) 



and supposing a, /3, y to be the angles which r forms with 

 the positive semiaxes, we have also 



Ax Ay r, As ,^ . 

 = cos a -^^ = cos p = cos y (2.) 



