and the Laws of Plane Waves proj)agated through them. 129 



Hence a function of (\, /x, v, (p, x, ^) will reproduce itself by transformation 

 of co-ordinates ; let the function be 



2V^P\+ Qix + Rv + 2F<j, + 2(?x + 2^?^^ 



I shall first prove the existence of three rectangular axes, for which this 

 function reduces to its first three terms. If the axes of co-ordinates be trans- 

 formed, and the coefficients of (0', x', ^') equated to zero, we obtain 



Pbc +-Qb'c' + Eb"c" + F{b'c" 4- b"c') + G(bc" + b"c) +H{bc' + b'c) = ; 

 Pac + Qa'c' + Ra"c" + F{a'c" + a"c') ^ G{ac" + a"c) + H{ac' + a'c) = ; 

 Pab + Qa'b' + Ra"b" + F{a'b" + a"b') + G{ab" + a"b) + H(ab' -f a'b) = 0. 



These equations will be satisfied by assuming for axes of co-ordinates the axes 

 of the ellipsoid 



Px" -t- Qif + i?--- + 2Fyz + 2Gx~ + 2Exy = 1. 



Hence, for these particular axes, 



2 F = P{u' - 4^7) -1- Q{v' - iay) + R(w' - iap). (32) 



Using this value of T^in equations (30), we obtain for the equations of motion, 



(33) 



(X, F, Z) denoting the same functions as in (27). If the function were 



' 2 F= PX' + QY' + RZ\ (34) 



equations (27) would be the same as (33). 



These equations are those used by Professor Mac Cullagh and by Mr. Green. 

 They denote transverse vibrations, and will give Fresnel's wave-surface for 

 plane waves, and also the vibrations parallel to the plane of polarization. Mr. 

 Green has deduced his equations from a homogeneous function of the second 

 order of (a, /3, 7, m, v, w), by restricting it so as to be capable of propagating only 



VOL. XXII. S 



