57 
1912-13.] Dr Muir on Axisymmetric Determinants. 
that is to say, into an aggregate of multiples of squares of linear functions 
of the variables : for example, 
ax 2 + by 2 + cz 2 + 2 dyz + 2 ezx + 2 fxy 
f e \ 2 ab - 
a a-K-y-f-z) + — 
a a J a 
, ad — ef 
y+ ^ z . 
abc + Zdef — . . . w2 
ab-f 2 
Finally, as the quadric is invariant to a variety of sets of interchanges, 
there must be a corresponding variety of sets of conditions : and, as these 
latter sets must be all coexistent, there follows an interesting theorem 
which we may formulate for ourselves thus : If the axisymmetric deter- 
minant | a 1 b 2 c 3 d 4 1 and its coaxial minors | a 1 b 2 c 3 1, | a 1 b 2 1, a 4 be positive, then 
all the other coaxial minors are positive also. 
Having had in the foregoing to transform 
| I I I a l C Z I | J | I I I ft l C 3 I I | j .... 
into 
a l | flq&gCg | , a 2 \ afb 2 c z d^ \ , .... 
Williamson proposed the general problem in the Educ. Times, where 
solutions by Townsend, Laverty, and himself duly appeared. Ohio’s more 
general result of 1853 was apparently unknown. 
Buchwald’s paper is closely on the lines of Williamson’s. 
Ritsert, E. (1872). 
[Die Herleitung der Determinante ftir den Inhalt des Dreiecks aus den 
drei Seiten. Zeitschrift f. Math. u. Phys., xvii. pp. 518-520.] 
This merely gives a variant on the usual way of deducing the four-line 
determinant in question from the determinant which involves the co- 
ordinates of the vertices. 
Cayley, A. (1874, April). 
[On the number of distinct terms in a symmetrical or partially 
symmetrical determinant. Monthly Notices R. Astron. Soc., xxxiv. 
pp. 303-307, 335 : or Collected Math. Papers, ix. pp. 185-190.] 
Denoting by cf>(m, n) the number of distinct terms in a determinant of 
the (m + n) th order having an m-line coaxial minor which is axisymmetric, 
