1899-1900.] Dr Muir on the Theory of Alternants. 
123 
These preliminaries having been given, the second proposition 
may now be proceeded with. It is — 
If <f> be any rational integral function of m + 1 variables , II their 
difference-product , and f be a function of the (n + l) th degree in 
one variable and be of the form (x - a 0 )(x -af ... (x- a n ), 
then when m;j>n the coefficient of t 0 ~ 1 t 1 ~ 1 . . . t m _1 in the expan- 
sion of 
n(t 0 ,ti, . . ., t m )<^>( tp.t 1 , . . t m ) 
f(t 0 )f(tl) .... f(t m ) 
It is easily seen that there is still an analogue when the point through 
which the parallels are drawn is inside the triangle : thus, corresponding 
to the diagram 
we have the identity 
(Kabcdefg) = _ (/ _ a) ^ _ a){/ _ b){g _ b)i 
and as (/- a)(g - a) = (/- a){g - a) \ 
•if- b){g-b) (f-D(g-b) 
( f-c)(g-c ) 
(f-d)(g-d)UCKf,9)CK9,f) 
{f-e){g-e) 
. (9-f)\ ' 
if ~9) • 1 
= f\f)f\9 ) t {-HKfMK m 3 
it follows that 
Ciabcdeftff .C&de) _ 
C 1 *{abcde).(i(cdefg) (Kf9)(Kf9) * 
It should be noticed, however, that the absolutely perfect geometrical 
analogue to Jacobi’s identity is got by taking a 
rectilineal figure of the form ABODE, where B 
AB = BC, CD = DE, B = C = D = 90°, and then 
equating the sum of the two parts got by joining 
CE to the sum of the two parts got by producing 
DE to meet AB in E. Further, the exact 
analogue to his proof would be to say that the 
rectangle BCDF contains all of the triangle ABC 
except the triangle AEF, and contains the 
triangle ODE in addition. 
