OF EVERY DEGREE OF TRANSPARENCY. 219 



the real part represents the averaged displacement [U] Ave , and the 

 coefficient of i the rate of increase of the same multiplied by a 

 constant factor. This bivector therefore represents the average 

 state of a small part of the field both with respect to position and 

 velocity. We may also say that the coefficient of i in W represents 

 the value of the averaged displacement [U] Ave at a time one-quarter 

 of a vibration earlier than the time principally considered. 



11. It may serve to fix our ideas to see how W is expressed as a 

 function of the time. We may evidently set 



r , , , , 



[U] Ave = A! cos --f A 2 sm t 



where A x and A 2 are vectors representing the amplitudes of the two 

 parts into which the vibration is resolved. Then 



P r ' T . 27T, 27T, 



^WUe= -Aism e+A,coBy*, 



and 



MAve- ^ [U] A ve = (A 1 - i A 2 )( C OS y t + l sin -j- 



that is, if we set A = A x i A 2 , 



(13) 

 In like manner we may obtain 



2fftt 



Q=ge p , (14) 



where g is a biscalar, or complex quantity of ordinary algebra. 

 Substituting these values in (12), and cancelling the common factor 

 containing the time, we have 



4-7T 2 



(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 numerical 

 quantities (viz., the three components of A 1? and the three of A^, 

 which may be called the six components of amplitude. The equation, 

 therefore, substantially represents the relations between the six com- 

 ponents of amplitude in different parts of the field. t The equation 

 is, however, not really different from (12), since A and g are only 

 particular values of W and Q. 



*The terms VQ> V<7 are allowed to remain in these equations, because the best 

 manner of eliminating them will depend somewhat upon our admission or rejection of 

 the solenoidal hypothesis. 



t The representation of the six components of amplitude by a single letter should not 

 be regarded as an analytical artifice. It only leaves undivided in our notation that 

 which is undivided in the nature of things. The separation of the six components of 

 amplitude is artificial, in that it introduces arbitrary elements into the discussion, viz. 

 the directions of the axes of the coordinates, and the zero of time. 



