( 2 09 ) 



VII. — The Automorphic Linear Transformation of a Quadric. 

 By Thomas Mum, LL.D. 



(Read April 5, 1897.) 



(1) It is well known that Cayley effected the transformation of 



Xl z+xJ+x*+ . . . +^ 2 into &'+&*+&"+ • • • £ 2 

 by introducing an intermediary set of variables 



01 > $2 > ^3 » • • • > 0« 



connected with each of the other sets by means of a linear substitution of a peculiar 

 type. The substitutions in fact were 



^11^1 + ^12^2 + ^13^3 + 

 V*l + *2202 + *2303 + 

 ^1^1 + ^2 + ^3 + 



• = *!*) 



■■x 2 I 





and 



J 



^lA + ^A + ^31^3+ ■ ' • =fl > | 

 ^12^1 + ^22^2 + ^32^3+ ■ • ■ = ^2 



^ 



^1 + ^2 + ^3+ • • • =?3 I 

 J . 



where the determinant of the first substitution is unit-axial and skew, and the deter- 

 minant of the second substitution is got from the preceding determinant by changing 

 rows into columns, and where, therefore, the number of arbitrary quantities introduced 

 by the two substitutions is only \n{n — 1 ). Cayley did not in any way indicate how 

 he was led to the substitutions. It has to be carefully noted, however, that when they 

 had been divined or devised, the essential difficulty of the problem had been overcome ; 

 all that remained was the simple algebraical process of eliminating B x , 2 , 3 , . . . and 

 so finding the expression for each of the x's in terms of the £'s. It is true that this 

 process does not take a simple form in the original memoir.* In later days, however, 

 he would doubtless have dismissed the matter in a couple of lines. For, writing the 

 two sets of substitutions in the form 



Cu 



hi 



hs ■ ■ ■ 



h 



hi 



hs • • • 



hi 



hi 



hi • • • 



(hi 



hi 



hi ■ ■ • 



hi 



hi 



hi • ■ • 



hz 



hs 



l S3 . . . 



)(01> 02' 03' •• •) = (*!> X 1: 



•). 



) (01 » 02 ' 03 ' •••) = (?!' £> ' £s> ■ ■ ■ ) ' 



* Cayley, A., "Surquelques proprietes des determinants gauches," Crelle's Journ., xxxii. pp. 119-123 ; or Collected 

 Math. Papers, i. pp. 332-336. 



VOL. XXXIX. PART I. (NO. 7). 2 I 



