III. On the General Equation of Surfaces of the Second Degree. 

 By Augustus De Morgan, of Trinity College. 



[Read Nov. 12, 1832.] 



The present investigations are a continuation of those upon lines 

 of the second degree, published in Vol. IV. Part I. of these Transactions. 

 I have omitted various algebraical developments, as unnecessary, and 

 tending to swell this communication to a length more than proportional 

 to its importance. 



As the theory of the reduction of oblique to rectangular co-ordinates 

 is a very necessary part of what follows, I proceed first to give the 

 equations which will be required under this head. Let x, y, %, be 

 oblique, and x', if, a' rectangular co-ordinates to the same point, with 

 a common origin. Let the angles made by the first system be 



A A A ^ 



y% = ?, %x = t), xy = ^, 



and let the rectangular and oblique co-ordinates be so related that 



AAA 

 COS xsd = a, COS yx' = /3, cos xyf = a', &c. ; 



whence the following equations: 



a/ = ax + fiy + yx, 



y' = a'x +, /3'y + y'z (1), 



S8' =a"x + fi"y + 7"i8; 



l = a'+a" + a'", COS ? = /37 + (i'y' + fi"y'\ 



l=l3f>+ fi'^+ fi'% COS t, =ya + y'a! + y"a" (2), 



1 = y + y^ + y% cos ^ = a/3 + a'/3' + a")8". 



