152 Proceedings of the Royal Society of Edinburgh. [Sess. 
and all that remains is the evaluation of the bracketed factor after the 
equivalents of P and N have been substituted therein. The final result is 
u u 
U \3 
U 2l 
u 22 
U 23 
U 2>\ 
U 2>2 
U SS 
• I ^h( 0'273 a 372^ 2 W 22( a 37l a l73) 2 U 3z( a lY<2 a 27l) 2 
+ (?/ 23 + %2K a 37l ~ a l73) a l72 — a 27l) 
+ (u 2l + ■W 13 )(a 1 y 2 - a 2 y 1 )(a. 2 y 3 - « 3 y 2 ) 
+ (W 12 + *%)(a 2 73 ~ a 37 2 )( a 37l “ a l73) 
The case where n — 4 is similarly dealt with ; but as it is necessarily more 
complicated, it is not carried quite so far, the cofactor of | u n u 22 u 33 u 44 ! 
being merely stated to be of the desired form and easily calculable. “ Sie 
hat aber zu viele Glieder, um sie berechnet hinzuschreiben.” The general 
proposition, as above given, is then formally enunciated. 
The other generalisation made of the case where n = 2 is to the effect 
that the product of an axisymmetric determinant by the square of any 
other determinant is expressible as an axisymmetric determinant. In 
connection with this the interesting point is the notation used for the 
elements of the product-determinants. Since the differential-quotient of 
2 { u n x i^ + ^22*^2 2 + • • • + 'U'nrfin "h 2 ^^ 2 + ... } , 
or F(aq, x 2i . . . x n ) say, 
with respect to x p is 
u lp x x + u 2p x 2 + . . . + U vp x p + . . . + U np x n 
the result of annexing q as a second suffix to the os’s in this may be suit- 
ably denoted by 
so that in accordance with this the product of 
U 11 U 12 * * 
. . U ln 
x i\ x 2] 
.... x nl 
W 21 ^22 * * 
. . u 2n 
and 
^12 ^02 
.... x n2 
u n\ U n2 • • 
• • \n 
= ^ \k 
x \n X 2n 
• • • • x nn 
F(*n) 
P (^2l) * • * 
P'(^nl) 
P 0*a2) 
F(z 22 ) . . . 
P'M 
P ( X ln) 
P ( X 2n) • • • 
“ Multiplicirt man diese Determinante nochmals mit der vorhergehenden 
und setzt : 
PpS ( X lq) "h (p^2q) • • • d" *®«pP ( fnq ) 5 
