61 
1912-13.] Dr Muir on Axisymmetric Determinants. 
coaxial primary minors. A perusal of the paper on the “ Method of Least 
Squares” will show how the expression originates* (see §§ 5, 6). When 
freed of awkward notations it is, in the case of the fourth order, 
1 1 
I «A C 3 I 
= (L 
I a A 
| a 
a i b 2^4 | 2 
I «A I * I a A C 3 I 
We may note for ourselves that the corresponding identity when axi- 
symmetry is not insisted on is 
| a i b 2 C 3 ( ^ , 4 i 
af) 2 c l 
a^d 1 _ \a l b±\\ a 1 d 2 | _ | a f) 2 c^ \ \ a Y b 2 d z \ 
CLi a 1 | a l b 2 | | a Y b 2 | | a A c s | 
and that a very instructive way of establishing it is to combine the first 
two terms on the right into one, then in similar manner combine the result 
thus obtained with the third term, and so on, the successive “ approxima- 
tions,” so to speak, being 
^ | a Y d A | | af) 2 d A | | af> 2 c % d± \ 
4 ’ «1 ’ vt a i b 2 I ’ I a l b 2 c 3 I ’ 
The series is thus seen to be one of those that close up telescopically. 
Glaisher, J. W. L. (1876). 
[Theorem relating to the differentiation of a symmetrical determinant. 
Quart. Journ. of Math., xiv. pp. 245-248.] 
The theorem is to the effect that, If u — l/A* where A stands for 
2«n 
a 12 
a ] 
a 2l 
2(X>22 
a< 
a 31 
a 32 
2a. 
^ rs ® sr j 
then any derivative of u with respect to one or more of the as is equal to 
any other such derivative having the same number of occurrences of each 
suffix-number. For example, 
_02__0__a_ LJLJLJL d A JL JL u 
bCL -^ 2 0^23 0ttg^ 0(^22 0^24 ^^33 ' 0^13 0^22 ^^24 
* Other papers on the same subject are — 
Geer, P. van, “ Over het gebruik van determinanten bij de methode der kleinste kwa- 
draten,” Nieuw Archief voor Wiskunde , i. (1875), pp. 179-188. 
Catalan, E., “ Remarques sur la theorie des moindres carr4s,” Mem. de VAcad. . . . de 
Belgique , xliii. (1878), pp. 24-33. 
