493 
1887.] Dr T. Muir on the Theory of Determinants. 
must be accompanied by two others, 
| x k y h z t | and | xpy h z k | , 
and the sum of these is 
I W | - | XhV&i | + | x hV£i i , 
i.e n 
I x hVh z i | • 
The final result is thus the sum of the resultants of the form 
c i 
I x kVi£i | where h<k<l, and 1 = 3, 4, . . . , n, 
the number of them, as we may see from two different standpoints, 
being 
J n(n - 1 ){n - 2) . 
Returning to the series of identities, 
^3 1 ^1^2 1 "b y 3 |^ 2 l d" % | ^ 1 ^/ 2 1 ~ \ x lV2 Z z\ ’ 
^4l2/i%l + y 4 l%l + = K V2h\’ 
(fee. (fee. 
which by addition give the result 
%xt\y x z 2 \ + %y%\z x x 2 \ + %z%\x Y y 2 \ = %\x x y^\, 
Binet next raises both sides of all of them to the second power, and 
obtains 
^\ x iU ^\ 2 = 2 « 2 :% i 2 2 .| 2 + % 2 5|^ 2 | 2 + 3 z 2 2 | aqy 2 | 2 > 
I 
+ 2Sjz2(iz 1 * 2 |.|*iy 2 l) + 2Sz*5(|* 1 ?/ 2 |.|?/ 1 2 2 |) i(xxxn.) 
+ 2S*2/S(|2/ 1 z 2 |.|z 1 a: 2 |)J 
Substituting for . . . . , their equivalents as given 
by the multiplication-theorem, he then deduces 
S|W3l 2 = ^ 2 % 2 ^ 2 d~ 2 %yz%zx%xy — ^ 2 (^2*) 2 | 
-%y\%zxf - %z\%xyf,) 
not failing to note that this is not a fresh result, but merely a case 
of the multiplication-theorem in which the factors are equal. 
By putting the right-hand member here into the form 
%y^{%^%x^ - {^yzf} + ^ 2 l^ 2 % 2 -(^y) 2 } 
- %x 2 {%y 2 %z 2 - (^yz) 2 } + T%yz\%zx%xy - %yz%x 2 } , 
there is next arrived at the first identity of the set 
