OF ARTS AND SCIENCES. 239 



This will evidently give a single finite value for 8, unless one of the 

 c's vanish. Take the most general case. Let y be in any direction, 

 such as 



y = — c?«i — <7«., — ka^. 

 Then 



S«jj' = — c?S«j" = -^ d, 



S«27 = -[- ^? ^ud S«g7 = -j- k, 

 5 = (^ + ~ + ^)(— ^'^1 — 9<h — k^h) + (^ + ^) ^«i -I- 



1 1 1 



= — ^ '^"l — ^ 5'«2 — -^ ^-^S- 



Then if one root, say Cj, vanishes, the centre is at an infinite distance 

 in the direction of «j, but at a finite distance in each of the otlier direc- 

 tions. Now let us substitute 



^1 = 0, 

 and 



J' = — t/wj — ga.^ — ha^ 



in the general equation of the second degree 



^QcpQ -f- 2 Sj'p =. c = qS"«jO -|- CgS'ag? -|~ ^s^'^zQ ~\~ 2 Sj'(). 

 We find 



c^'^a^o -\- CgS^ttgO — 2 c?S«j(> — 2 gSa.^o — 2 ^'SoCgp =: c. 



We found that —g and —k were the distances of the centre in the 



directions of «., and «„. To bring the origin into a line with the cen- 

 tre in these two directions, substitute 



P = P — "^ «2 — 7^ «3» 



and the equation takes the form 



c^S^UgO -\- CjS^rtgO — 2 c?S«jp = c. 



This may be still farther simplified, by taking for origin that point 

 where a^ meets the surface. Let 



