IV. AXES OF CENTRAL QUADRIC. 



571 



We will now transform the equation of the quadric l) to a new set 

 of axes coinciding in direction with its principal axes. Let the new 

 coordinates be x\ y\ #', and let the direction cosines of the angles made 

 by the new with the old axes be given in the table below. 



The equations of transformation of coordinates are then 



x = a^x + ft?/ -f yi s, 

 18) y 1 --= a^x -f ft?/ -f 7 2 2, 



x = a^x' 4- 



19) y = (l lX ' + 



' -f 

 Now using equations 19), we obtain 



ft 



which in virtue of equations 15) is equal to 



*!!#' + ^22/' 

 In like manner 



Multiplying respectively by #, /, ^ and adding, we obtain 



Ax 2 + By* + <? 2 + 2D/^ + 2Ezx 

 pi l^x'fax + ft?/ + 7^) + ^fax -f ft/ + 7 2 *) + 



Consequently the equation of the quadric referred to its principal axes is 



20) AX 2 + V 2 -fM' 2=1 , 



and the three roots of the cubic are equal to the squares of the reciprocals 

 of the lengths of the semi -axes. Accordingly in order to find the equation 

 referred to the axes it is not necessary to solve the linear equations 5), 

 but only to solve the cubic 6). 



