in Media of every degree of Transparency. 115 
averaged displacement [U]aye, and the coefficient of ¢ the rate 
of increase of the same multiplied by a constant factor. This 
bi-vector therefore represents the average state of a small part 
- It may serve to fix our ideas to see how W is expresse 
as a function of the time. We may evidently set 
2a prise 
[Ulave = A, cos ~~ ¢ + A, sin ie t 
where A, and A, are vectors representing the amplitudes of 
the two parts into which the vibration is resolved. Then _ 
, aed 27 
£ [U]ave = — A, sin > t + A, cos ry t, 
and 
* 27 peg Es 
[U]ave watt. z” [U]ave = (A, — z A,) (cos > t+1 ae t); 
that is, if we set A=A,—¢A,, 
amt 
W = A e Pp S (13) 
In like manner we may obtain 
Q =J9 eP, (14) 
where g is a bi-scalar, or complex quantity of ordinary algebra. 
Substituting these values in (12), and cancelling the common 
actor containing the time, we have 
=e Pot A- rg = OA. (15) 
Our equation is thus reduced to one between A and g, and may 
easily be reduced to one in A alone.* Now A represents six 
humerical quantities, (viz: the three components of A,, and the 
three of A,), which may be called the six components of ampli- 
tude. The equation, therefore, substantially represents the 
relations between the six components of amplitude in different 
parts of the field.+ The equation is, however, not really 
* The terms vQ, vq are allowed to remain in these equations, because the b 
— of eliminating them will depend somewhat upon our admission or rejec- 
“on of the solenoidal hypothesis. : we 
€ representation of the six components of amplitude by a single letter 
notation that which is undivided in the nature of things. e separation of th 
SIX Components of amplitude is artificial, in that it introduces arbitrary — 
into the discussion, viz: the directions of the axes of the codrdinates, 
Zero of time, 
