390 Dr. Oliver Lodge on Opacity. 



The slip was naturally due to a consideration of the 

 extreme frequency of light vibrations ; but attention to the 

 more complete expression for the solution of the same differ- 

 ential equation, given in 1887 by Mr. Heaviside and quoted 

 in the note to this Society above referred to, puts the matter 

 in a proper position. Referring to his ( Electrical Papers,' 

 vol. ii. p. 422, he writes down the general value of the 

 coefficient of absorption as follows (translating into our 

 notation) 



rsMH&JT-'}' 



without regard to whether the conductivity of the medium is 

 large or small ; where v is the undamped or true velocity of 

 wave propagation in the medium ({aK)~K 



Of course Maxwell could have got this expression in an instant 

 by extracting the square root of the quantity Q, the coefficient 

 of F in equation (o f ) written above. I do not suppose that 

 there is anything of the slightest interest from the mathe- 

 matician's point of view, the interest lies in the physical 

 application ; but as this is not a mathematical Society it is 

 permissible, and I believe proper, to indicate steps for the 

 working out of the general solution of equation (3) by extract- 

 ing the square root of the complex quantity Q. 



The equation is 



and the solution is 

 where 



Q = ^/-^K/ + ^^ =* + ;/? say. 



Squaring we get, just as Maxwell did, 



Squaring again and adding 



(a* + f3 2 ) 2 = (a 2 - /3 2 ) 2 + 4a 2 /3 2 = fi 2 Ky + ]^fjpt 



I 



wherefore 



^-v{. + (^)*}! 



2/ 3.= P K i ,.{ v /(l+(i^)*) + l}, . . (4) 



