514 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



Let x'y be co-ordinates referred to two new axes (a'/3') in the plane of xy, making respectively 

 angles 45" and 90' + 45" with the axis of x ; then 



Sx = —7- (Sx - By), Sy = —7= {Sx + ^y'), 

 \/2 \/2 



a = 4= («' - (3'), /3 = 4= («' + ^')- 



Making these substitutions, we find 



^" = * d^y ^«'7i (? - ") - /3' Ti^" - ?) - ^n (^^''^ - ^y')- 



Hence, by what has been already proved, the force brought into play will be 



7. We may now write down the symbol of the whole force brought into play by the dis- 

 arrano-ement represented by the expression (l), neglecting terms beyond tliose of the second order. 

 It will be as follows, 



'-"UhigH^)]' 



+ (A-B) 



dx' dy dz^ 



d- _„ , d" . „„ rf- 



+ 



d^y^^^^'-^^^d^Jy^^f'O^j^Mt^y^)] 



The coefficients of a, /3, 7 in this expression are the well-known differential formulae for 

 the three forces (parallel to the three axes) brought into play by the displacements ^, rj, ^. 

 The part of F which is multiplied by J - B, may be put in the form 

 ( d d d\ id^ dti d^\ 



\ dx dy dxl \dx dy dzl 



Hence, the equation of motion of the medium (which includes the three ordinary equations) 

 assumes the following form, 



dt- \\dwl \dyl \dzl j ^ ' \ dx ^ dy ^ dzJ [dx dy dz) 



8. This equation may be put in a remarkably simple form by the use of the notation I\u'.u. 

 Let us assume the symbol 3B to represent the operation 



d d d 



"T- + P:7- + 7T-. 

 dx dy dz 



then, since u = a^ + /3f) + 7^, we have 



dx dy dz 



._ d? d,, dX 



