532 



Proceedings of the Royal Irish Academy. 



XXVIII. — Note on the Chaeactee of the Ltneae Teansfoemation 



WHICH COEEESPONDS TO THE DISPLACEMENT OF A ElGID SySTEM IK 



EiiiPTic Space. By R. S. Ball, LL.D., F.R.S. 

 [Read, November 9, 1885.] 

 I WEiXK the general linear transformation 



3/i = 11 a^i + 12 Xz-^ 13 Tj + 14 a?!, 



t/s = 31 a;i + 32 x^ + 33 Xs + 34 X4,, 

 1/i = 41 ;j;i + 42 x^ + 43 :i;3 + 44 x. 



It is known that this is too general to denote the displacement of 

 a rigid system in elliptic space. It must be specialized so that a cer- 

 tain quadric surface (as a matter of fact there is a family of quadric 

 surfaces) shall be displaced upon itself. This implies one condition, 

 but only one, to be satisfied by the sixteen coefficients 11, 12, &c. The 

 algebraical character of this condition has not, so far as I know, been 

 hitherto pointed out. It is of an interesting nature, though I have not 

 thought it necessary to attempt the portentous task of developing its 

 actual expression. "We shall first enunciate the theorem, and then 

 give the demonstration. 



Form the biquadratic equation in p which is produced by the 

 development of the determinant 



11 



-p, 



12 , 



13 



» 



14 



2! 



> 



22 -p, 



23 



) 



24 



sT 



> 



32 , 



33- 



■P, 



34 



il 



) 



42 , 



43 



1 



44-p 



= 0. 



Let a, p, y, S be the four roots of this equation ; then the sym- 

 metric function 



{afS - yS) (ay - /?S) (aS - /?y) 



having been formed gives, when equated to zero, the required condi- 

 tion. 



