292 The Rev. Professor O'Brien on the Laws 



brations, will be (see Cambridge Transactions, vol. vii. p. 

 409) *, 



di^ ~ ^ \dx^ ^ dxf "*" dzV dt ^df^' K . (3.) 



and two similar equations for >) and ^. J 



For normal vibrations we have three equations of exactly 

 the same form as these, differing only in having the constant 

 A in place of B. 



{N.B. By transversal vibrations, we mean those which sa- 

 tisfy the condition 



d^ d r dK 

 dx dy dz 

 and by normal vibrations tliose which satisfy the three condi- 

 tions 



d^ _ dvj dfi _ d}^ dt, _d^ 

 dy d x' d z dy* dx~ dz' 

 These definitions of transversal and normal vibrations apply 

 equally, whetiier ^, >), ^ be imaginary or not.} 

 If, in the equations (3.), we put 



P B 



-C, -ii^ = B', 





1+Q ' 1 + Q 

 vee have 



dt^ ' ^ dt Vr/^2 ' dy"^ ' dz'^J' J,. . (4.) 



and two similar equations for >) and ^. 



These three equations are satisfied by the following imaginary 

 values.of ^, ij, ?, viz. 



^ =z a'u, Yi = b'u, ^:= d u, 

 where ^ = g{«^-*'(/^+?'«+*'^)}^/^, 



and consequently by substitution, 



„2_c'w^iri = B'F2 (5,) 



Also, since the vibrations are transversal, we have, by the de- 

 finition above-given, 



p' a! + q'b'+ s' c' = 0. 

 From equation (5.) it is manifest that, supposing n real, k' 

 is imaginary. By superposing two proper sets of imaginary 

 solutions thus obtained, we shall obtain real integrals of the 

 equations of motion. 



* To conform to the most usual notation, we have in the present papers 

 represented the displacements by the letters i, n, ^, instead of «, /3, y. 



