824: Prof. J. W. Nicliolson on Uniform Rotation, Principle 

 Substituting in the set of equations typified by (4), 



4z7T6) 



,3/' 



dt' -Yd6' + c 2 3W 6 



■kir 





9a' 3c' 



M*V) 



1 



}•■ (7) 



, ,9A' 3 ,,, ,. /3 , ^»'' 2 3\ , 

 w 37 = 3V (6 ' ) _ W + ~W 3?> 



A variation in the usual procedure for Cartesian coordi- 

 nates must now be introduced, e being variable. Write 

 p 2 = er" 2 } so that 



p 2 =r'V(l-«V 2 /C 2 ), .... (8) 

 and conversely, 



r' 2 = p 2 /(l + »y/C 2 ), e-l + ^/C, . . (9) 

 so that 



dp/d* J =pP-lr' z , pdp\r'dr=e\ . . . (10) 



The length p is a radius vector in a certain distorted space 

 which may be derived from the original space by a non- 

 uniform extension perpendicular to the axis, around which 

 it is symmetrical. If dA. is a polar element of area in this 

 space between two radii of length p enclosing an angle d6', 

 then dA = j>p q d6'. Let the time variable be changed from 

 t' to t'\ where 



£" = *'- 2a>A/C 2 (11) 



Then 'dj'dt' becomes 3/d£", and 'dj'dO' -f ecoG~ 2 r 2 'd/'dt / becomes 

 'd/'dO'. Accordingly, 



, ,3/' 3</ 3 , , 



^K=* a ' 



,3_A' 



irrer' ~. = ~, (6V) 



3*' 



3 



Zr 



aw 



Y, 



3_a' 



30' 



(12) 



with similar equations for the time rate of the magnetic 

 vector. 



These equations are of the same form as those from which 

 we started, except for the factors e on the left. Since these 

 differ from unity by amounts of the second order (for proper 

 values of r'), the correspondence between the fixed and 

 moving systems follows in the usual way to the first order. 



The difference between r and p is only of the second order, 



