178 PROCEEDINGS OP THE AMERICAN ACADEMY. 



VI. 



ON THE AUTOMORPHIC LINEAR TRANSFORMATION 

 OF A BILINEAR FORM. 



By Henry Taber. 



Presented January 10, 1894. 



In the Philosophical Transactions for 1858, Cayley gave the fol- 

 lowing determination of the general automorphic linear transformation 

 of the alternate bilinear form with cogrediant variables 



A A 



(n)(x u x 2 , . . . )(yi, y 8 , . . . ), 

 namely, 



(Q _ Y )-i (Q + Y), 



in which Y denotes an arbitrary symmetric linear transformation. 



As shown by Cayley, this is equivalent to the determination of the 

 general solution of the matrical equation <£ 12 cf> = 12, in which 12 is a 

 known skew symmetric matrix, and <£ denotes the transverse of </>. 

 Cayley's solution of this equation fails for those matrices </> of which 

 —1 is a latent root. 



If 12 is real, and is both skew symmetric and orthogonal (i. e. if 

 O -1 = 12 = — £2), the product of two of Cayley's expressions gives 

 every real solution of this equation. 



Thus, let <f> be any real solution of the equation <£ 12 <f> = 12, of 

 which —1 is a latent root. A polynomial <t> = /(<£) in integer powers 

 of <£ with real coefficients can be formed, containing every factor in 

 the identical equation to </>, except <£ + 1, and such that 



4> 2 = <&, $ 12 = 12 $.* 



* The latent roots of </> being ±1 of multiplicity m and n respectively, 

 9r> 9~ l > each of multiplicity p r , for r = 1,2, ... i, the identical equation to <p is 



and if 



F r (*) = [(* + 1)» - (g+ l )*Y* [(* + 1)" - (9~ l + I)"?*, 



