(97) 
NOTE ON THE RESOLVABILITY OF THE MINOKS OF A 
COMPOUND DETERMINANT. 
By Sir Thomas Muir, LL.D. 
(1) The theorem which is usually made the basis of investigation on 
this subject is that exemplified by Spottiswoode * in 1853 and formally 
enunciated and established by Franke and Borchardtf in 1862. It is in 
effect that any Jc-line minor of the mt^ compound of |ai„ is equal to 
K being the complex entary of the corresponding minor in the {n—my^ com- 
pound. The theorem is of course directly useful for the end in view when 
h > {n — in other cases it may turn out indirectly useful, but only 
because the said corresponding minor occasionally lends itself more readily 
to decomposition than the minor actually set. 
(2) There is one case, however, in which it is of no avail at all; 
namely, where 
h — (71 — 1),„ and n = 2m, 
for then it takes an illusory form, all the information given by it being that 
the minor in question is equal to itself. The existence of this case makes 
the general problem difficult of treatment ; and it is consequently desirable 
for future work to know how the question of resolvability stands in regard 
to it. The instance of it where 
n = 4, 7n = 2, k = 'S, 
is the simplest, but, as there are then 400 minors to be adjudicated on, 
considerable instruction necessarily results from examination of it. If the 
basic determinant be taken in the form 
the 400 in question are the three-line minors of 
\a-J).2\ kjc.^l . . . \c^dc>\ 
\a^b.^\ \a^c.^\ . . . \c^ds\ 
\a.^b^\ \a^c^\ . . . \c.d^\ . 
* ' Crelle's Journ.,' li, pp. 366-368. 
t ' Crelle's Journ.,' Ixi, pp. 350-358. 
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