60 
Proceedings of the Royal Society of Edinburgh. [Sess. 
row of the adjugate with M must also vanish, or those which are in the 
same column. As a corollary it is added that when in addition A is 
axisymmetric and M is coaxial, there is no alternative. 
Following on this is a proposition less readily acceptable, namely : If 
an n-line axisymmetric determinant and n-2 of its primary coaxial 
minors simultaneously vanish, then all the other primary minors vanish 
also. It is at once seen that after the application of the preceding corollary 
the only elements of the adjugate that require consideration are those of 
the last two-line coaxial minor; in other words, that the adjugate must be 
of the form 
0 
0 . . 
. . 0 
0 
0 
0 
0 . . 
, . 0 
0 
0 
0 
0 . 
. . 0 
0 
0 
0 
0 . 
. . 0 
X 
Y 
0 
0 . 
. . 0 
Y 
z 
Further, it may be agreed that the data give us 
xX + yY = 0) 
yY + zZ = 0 V 
XZ - Y 2 = 0 ) ; 
but the deduction made therefrom, namely, that 
X = Y = Z = 0 , 
is clearly not the only one. A test case that we may put to ourselves is 
abb 
bee 
bee. 
Finally, Seeliger considers the primary minors of the product-deter- 
minant, showing that they must all vanish if the one factor and its primary 
minors vanish and the other factor does not vanish. 
Glaisher, J. W. L. (1874, 1878). 
[On the solution of the equations in the method of least squares. 
Monthly Notices R. Astron. Soc., xxxiv. (1874), pp. 311-334.] 
[Questions 4418, 5530. Educ. Times, xxvii. p. 69 : xxxi. p. 21. Solu- 
tion by E. J. Nanson in Math, from Educ. Times (2), ii. pp. 95-96.] 
What is given in the Educ. Times is an expression in the form of a 
series for d’Arrest’s ratio of an axisymmetric determinant to one of its 
