PART II. POLAR MAGNETIC PHENOMENA AND TERRELLA EXPERIMENTS. CHAP. VI. 68 1 



hence 



r 2 /' i 2 iAf 

 C = -f- v n and a = . 



'"o r o 



If these values are introduced into (IV), we obtain 



(' 2 V , 



' 



If = x, - = A- O is introduced, the equation changes to 

 '' r o 



(A) AW*** (* - x,Y + 2 (.v - A O ) - ; 0. 



If .v is to be a double root in this equation, the equation 



(B) KM*x (x - -v c )- + t?M*x* (x - x ) + /< = 

 uist take place at the same time. The last equation may also be written 



ncl as / is negative, it will at once be seen that the double root must be positive. But we can prove 

 lat it must also be <^ A - O ; for from (A) and (B) we obtain 



KM- (2.i- .*) x (x x g )- , 



As vl is positive, it is evident that this equation cannot take place unless x<^x u . We see then, 

 lat if there is any double root at all in (A), it is positive and<^A" , and a double root in equation (IV) 

 necessarily positive and ^> r a , as it should be. 



In order to find the condition for the double root and its value, we must eliminate x from (A) 

 id (B). This is easily done in the following manner. 



By multiplying by x(x x a ) on both sides in (B), we obtain, on substituting the value of A" 2 (A- x a )~ 

 om (A), 



(2 * x a ) (vl 2/i (x *.)) + / (x *) = , 



; 



(C) 3 fix (x - or.) -f 2/<A- (A- - A-,,) + (2A- - A O ) v\ = 0. 



If we multiply here by 2x x a , and substitute the value of A- (A- x )(2x A- O ) from (B), we obtain 



Alii 



(D) 4 (ux + vl) x (x - x.) - (2 ft (x - x tt ) - vl) xl + . a ^ = . 



Then when x(x A' O ) is eliminated from (C) and (D), we obtain 



4 (/'* + V D ( 2 /"*o (x x n ) + (2 x A- O ) vl) + 3/i ( w ^ 2 (2/ (x A- O ) vl) xl} = 0, 

 hence we obtain 



x = 



