740 
MR. .7. LARMOR ON A DYNAMICAL THEORY OF 
any displacement which does not involve rotation, therefore such that the work done 
by any displacement whatever is of the form 
|(L8/+ IV% + N8A)f/T 
or 
together with possible surface-integral terms. Integration by parts leads to the 
expression 
This expression must be the same as the one derived by integration by parts in the 
\isual manner from the variation of the potential energy 8 |W(:Zt, where W is now of 
the second degree in spacial differential coefficients, of various orders, of (^, r], Q. 
The result, as far as the volume integral is concerned, will be the same as if the 
sjnnbols of differentiation cl/dx, d/dy, d/dz were dissociated from f, y, ^ and treated 
like symbois of quantity, after the sign of each has been changed, so that for example 
d^jdy d^yjdx^ is to be taken the same as — d/dy d^/dx'^ ; the function W may thus 
be reiDlaced for this purpose by 
W' = Af + + 2-Dy^ -f + 2F^^, 
where A, B, C, D, E, F are functions of d/dx, d/dy, d/dz. 
We shall then have 
On comparing these expressions there results 
/(m _ (m (VL _ l± ^ ±\ Ty / 
\ dy dz ’ dz dx ’ dx dy) ’ d^j 
Hence 
I d \ d\Y' / d \ dW' . fd\ d\Y' _ 
\dx) d^ '^[dy) dy d^ ~ ^ 
identically, where the differential operators in brackets are to be treated as if tliey 
were symbols of quantity. The vanishing of this expression, for all values of y, 1, 
involves three conditions between A, B, , . ., one of which may be stated in the form 
that the quadratic expression W' is the product of two linear factors; these are in 
fact the general analytical conditions that a medium shall not propagate waves of 
compression involving sensible amounts of energy. 
