110 SOME CURIOUS SIMPLE EQUATIONS. 
gether, almost haphazzard, three equations, apparently 
independent of each other, I found that they would not 
solve. The following is substantially the set of equa- 
tions which I happened on: 
(No. 1.) v+2y+32z=6 
a+4y+7z=12 
w+3y+52=9 
These are apparently independent but on trial it will 
be found that when reduced they give two identical 
equations. 
As some other teacher may happen on a similar set 
and be perplexed, for, so far as I know, no one has ever 
noticed, much less explained them, I propose in this 
paper to supply the deficiency, first giving by way of 
illustration several sets possessing the same curious pro- 
perties. 
(No. 2.) L—-Y-—W=4 
2%—5y—60=b 
18z—3y—2w=c 
(No. 3.) st+y—w=r 
' Ww 
(No. 4.) | w—y+3z—w=a4 
These will suffice. They can be indefinitely increased, 
_ and any number of letters may be used. 
sO 
