THOMAS MUIR ON BIPARTITE FUNCTIONS. 



481 



If, now, in this generalisation we make A axisymmetric, put £ n, £=£', >/, <£', 

 and consequently put A 2 = Ad we have the theorem that the discriminant of 

 the quadric resulting from a linear substitution performed on a given quadric is 

 equal to the discriminant of the original multiplied by the square of the 

 modulus of substitution.* 



* After the theory of this new class of functions had been worked out under a temporary designa- 

 tion of my own, I got the Philosophical Transactions for 1858, in consequence of a communication on 

 another matter from Professor Tait, in order to read Professor Cayley's Memoir on Matrices ; and there 

 found, immediately following the said memoir, another, " On the Automorphic Linear Transformation of 

 a Bipartite Quadric Function." This quadric function I saw at the first glance was a member of the 

 class I had been dealing with — viz., that of the third degree. This led me to discard the name I had 

 been employing, and to adopt bipartite instead. Professor Cayley gives the above extension of the 

 theorem regarding the invariance of the discriminant of a quadric, but without proof, and not as if 

 looking at it from that point of view. I think, however, I am correct in saying that this is the only 

 point in which my paper has been anticipated. Professor Cayley's notation for the bipartite we have 

 used above is 



( «i « 2 a u \ x y z I x ' v' *' ) 

 i h -i b i ! 



which does not, I think, bear on the face of it the exact nature of the two-sidedness of a bipartite of 

 the third degree ; that is to say, it does not imply, as 



X 



y 



2 



H 



a a 



a 3 



\ 



h 



h 



c i 



C 2 



c s 



does, that the function is equal to 



( (a^' + b x y' + c^z'yx 



either < + (a. 2 x' + b 2 y' + c^)y 



( + (a 3 x + h z y + r ? z')z 



t (rtj-e + a 2 y + a z z)x' 

 I + (\x + btf + b s z)y 

 f + (c^x + c$ + c z z)z' . 



It may be of interest, as another evidence of the usefulness of bipartites, to remark here that the 

 " Memoir on Matrices " came opportunely for another reason. The new instrument I had got hold of 

 seemed as if specially devised for dealing with matrices, and I immediately succeeded in proving 

 Cayley's great theorem that, if m be a matrix, the equation — 



is satisfied by 



a - u 



b 



c 



d 



e -m 



f 



9 



h 



k -it 





(a 



b c ) 



M 



= d 



e A 





'J 



h k ! 



= 



This proof, with its accessories, has been communicated to the Mathematical Society of London. 

 VOL. XXXII. PART III. 4 I 



