76 



Mr. CAYLEY, on THE THEORY OF DETERMINANTS. 



An equation, particular cases of which are of very frequent occurrence, e.g. in the investi- 

 gations on the forms of numbers in Gauss' Disqidsitiones Arithmetica, in Lagrange's Determi- 

 nation of the Elements of a Comifs Orbit, S^c. I have applied it in the Cambridge Mathe- 

 matical Journal to Carnot's problem, of finding the relation between the distances of five points 

 in space and to another geometrical problem. With respect to the notation of the second 

 section, this is so fully explained there, as to render it unnecessary to say any thing further 

 about it at present. 



{ I. On the properties of certain determinants, considered as Derivational Functions. 

 Consider the function 



U^x{al +(in + .••)+ (•)• 



X!'ial + ji'ri + ...) + 



{n lines, and n terms in each line). 

 And suppose 



KU = 



a, /3 . 

 a', d' . 



.(2). 



(The single letter k being employed instead of KU, in cases where the quantity (KU), rather 

 than the functional symbol K, is being considered). 



a{c + a'x' + .., bx + b'.v' + . . , .. (3). 



iif+sij+.., a , /3 ,.. 



FU=- 



k'^+ s'>; + 



'3U = - 



nx + R a- + . . , 



a'^+ Ti'tl+ .. , 



sx + s a; + ■ 

 /3' 



(*)■ 



The symbols K, F, 1 possess properties which it is the object of this section to investigate. 

 Let A, B, .. , A', B', .. , ■■ be given by the equations : 



A = 



A'=^ 



(5). 



(The upper or lower signs according as (w) is odd or even). 

 These quantities satisfy tlie double series of equations, 



^a +B/3 +.. = K, (6). 



Aa' + Bfi' + .. = 0, 



A'a+ B'I3 + ..=0, 

 A'a:+ B'ji'+ .. = K, 



&c. 



