1902-3.] Dr Muir on Axisymmetric Determinants. 
555 
The Theory of Axisymmetric Determinants in the 
Historical Order of Development up to 1841. By 
Thomas Muir, LL.D. 
(MS. received July 13, 1903. Read July 20, 1903.) 
Attention has already been drawn in Part I. of this history * to 
certain identities of Lagrange’s which might possibly be viewed as 
contributions to the theory of determinants. Among these were 
the following published in 1773 : — 
( xyz " + yz'x" 4- zx'y" - xzy" - yx'z" - zi/x") 2 
= ( x 2 + y 2 + z 2 )(x 2 + y' 2 + z 2 )(x" 2 + y" 2 -t- z" 2 ) 
+ 2(xx + yy' 4- zz){xx" + yy" + zz"){x'x" + y'y" + zz") 
- ( x 2 + y 2 + z 2 )(xx" + yy" + zz") 2 
- ( x ' 2 + ij 2 + z >2 )(xx" + yy" + zz") 2 
- (x 2 + y" 2 + z" 2 )(xx + yy' + zz') 2 ; 
(y'z" — y"z') 2 + (zx" - z'x') 2 + ( x'y " - x"y') 2 
= (x' 2 + y" 2, + z' 2 )(x" 2 + y" 2 + z" 2 ) - ix'x" + y'y" + zz') 2 ; 
and 
(j?r - q 2 )(Mn - Nm) 2 = (pM 2 + 2gMm + rm 2 )(^N 2 + 2 qNn + rn 2 ) 
- (pMN + qMn + qNm + rmn) 2 . 
Pour of the expressions here occurring would doubtless at a 
later date have been viewed as axisymmetric determinants, and in 
Cayley’s notation of 1841 would have been written 
x 2 + y 2 +z 2 xx -Vyy +zz 
xx + yy' + zz x' 2 + y' 2 + z' 2 
xx" + yy" + zz" xx' + y'y" + zz' 
xx -ryy + zz 
xx' + y'y" + zz" 
x" 2 +y" 2 +z" 2 
, etc, | 
but a reference to the original papers, already described, will make 
it almost perfectly certain that Lagrange did not view them in 
this light. 
The like is true of Gauss (1801) who discovered the next case 
of the third of the preceding identities. 
* Proc. Roy. Soc. Edinburgh , xiii. pp. 577-585. 
