OP ARTS AND SCIENCES. 225 



has been solved, and the centre found. Our equation then assumes 

 the form 



SgcpQ = — S8q)d — 2 Syd -{- c, 



and we may write it SgqjQ = c. 



By this process we have destroyed the three arbitrary constants 

 involved in y, and left only the six belonging to the self-conjugate gi^* 

 (Ham. Elem., § 358). This is precisely what should happen, for the 

 general equation of the second degree in Cartesian co-ordinates con- 

 tains nine arbitrary constants, while by taking the centre as origin 

 three of them are lost. 



If in the transformed equation 



Soqp(> = c, 



the constant vanishes, the equation represents a cone, since we may 

 give any value to the tensor of q, as the equation is homogeneous. 

 This case also I shall consider later. If the constant term does not 

 vanish, we can divide by it, and get our equation in the more con- 

 venient form 



Sgcp'g = 1, 



c disappearing into the new self-conjugate function (f'g. 

 If we differentiate 



SgqjQ = 1, 



we find (Tait, §§ 132, 251 c) 



SQcpdg -j- SdQq)Q = ; 



and since qig is self-conjugate, 



Sdgcf'Q = 8Q(j)dQ, 



or SdgqjQ = 0. 



VOL. XIII. (N. S. V.) 



