The So-called Vahlen Relations between the Minors of a Matrix. 699 
determinants of a matrix are but opportune transformations of the 
formula A." 
(8) With this pronouncement most mathematicians would agree. 
Unfortunately it does not appear to have led in twenty years to any better 
understanding of the subject : and one is thus induced to inquire on what 
grounds the application of Bazin's theorem to the matter can be viewed as 
an " opportune transformation." What advantage, for example, is there to be 
got by multiplying each of the (m)^ minors of the given array by the {n- 1)'* 
power of the first of them, as Vahlen directs, rather than by the first power 
of the said minor m accordance with Sylvester's theorem ? For one thing 
the sifting of the results into 
l-\-]c(m. — h) nugatory 
and (m) — 1 — ^ (m — k) effective 
cannot be the reason, because the same sifting takes place in both cases. 
Thus, the array being 
ai &i Ci c^j e^/i 
a., h,y C.2 d.y e.j/2 
ttg &3 C3 63/3 
we multiply, according to Vahlen, each of the 20 minors by \aib,)C^\^, 
finding 10 cases nugatory, 9 cases in which the result has to be simplified 
by the removal of the first power of \ciiboC^\, and only 1 case, namely, 
Id^e.jfg^.la^h^c.^l^ — \d-J).:>c^\ la^c^y'sl |^i^2^3l 
le^feg^sl 1^162^3' '(^i^o^sl 
I/1V3I l%/*2^3l 1^1 V3I 
which is considered to be useful as found ; in other words, if we arrange the 
20 minors in dictionary order and denote them by their order numbers, 
Vahlen' s result is 
8-1 = 
6-2 
- 5-3 
141 = 
12-2 
- 11-3 
91 = 
7-2 
- 5-4 
151 = 
13-2 
- 11-4 
101 = 
7-3 
- 6-4 
161 = 
13-3 
- 12-4 
17-1 = 12-5 - 11-6 
181 = 13-5 - 11-7 
19-1 = 13-6 - 12-7 
20-r 
11 5 
12 6 
13 7 
the nugatory cases being those whose left-hand members are- 
l•l^ 2-r, . . . , 7-r, 11 r, 12-p, i3-p. 
