CHARLES B. WARRING. bla 
It will be seen from the number of such equations that 
can be made that it may easily occur to a teacher in 
giving examples off-hand, to hit upon some of this kind. 
On examination it will be seen that the co-efficients in 
each equation of No. 1 are in arithmetical progression. 
The same is true of each of the other sets but the pro- 
gression is not apparent. It was obtained thus: I wrote 
r+94+3(y)=m 
2455) )=9 
32+) )=p 
Here the co-efficients are in arithmetic progression. 
We have only to remove the brackets and clear of 
fractions and then we have set No. 2. 
As to No. 3: I wrote, 
+22) x3) =a 
x+0(%)-1G)=¢ 
oa) 2 nel ~)=P 
and by the same process, a get me set. 
No. 4 was formed in like manner. In no case can 
equations, the co-efficients of whose terms form, in each 
equation, an ceiniaateipaee progression be solved. 
Proof. 
Let v+(1+a)y+(1+2a)z=¢ 
t+(14na)y+(1+2n4a)z=9q’ 
r+(1in’a)y+(1+2n’/a)z=q" 
be three general equations the co-efficients of which are 
in arithmetical progressions. 
By subtraction we have 
(na— a)y+(2na — 2a)z=9'— g 
(n’a— na)y+(2n/a — 2na)\z=q'—q’ 
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