ME. A. CAYLEY ON TSCHIENHAUSEN’S TEANSEOEMATION. 
569 
y- 
15, 
-26, 
-16, 
- 6, 
- 1 
1, 
^-10, 
-16, 
- 6, 
- 1 
1, 
6, 
y ^ 
- 6, 
- 1 
1, 
6, 
16, 
- 1 
1, 
6, 
16, 
16, 
2/+15 
and the values of the ten divisions respectively are 
y\ y\ f ^ f ^ y ^ 
6, 
-288, 
+ 4608, 
-24576 
1 
16, 
-576, 
+ 6144, 
-16384 
2 
26, 
-544, 
+ 3584, 
-24576 
3 
1, 0, -96, 
0 , 
-28672, 
0 
4 
0 , 
0 
5 
0 , 
0 
6 
26, 
+544, 
+ 3584, 
+24576 
7 
0 , 
0 
8 
16, 
+576, 
+ 6144, 
+16384 
9 
6, 
+288, 
+ 4608, 
+24576 
10 
1 , 0 , 0 , 0 , 0 , 0 
A verification similar to this was in fact employed at each step of the calculation of 
a division: viz. in forming a product such as (>tX+jM/Y+&c.)(X'X+jOt/'Y+&c.), where 
>v, fjtj, See., X', (jJ , .. See. are numerical coefficients, and X, Y, See. are monomial products 
of a, h, c, d, e, f and B, C, D, E, the sum of the numerical coefficients of the product is 
(X + + &c. ) (>i.' + p' + &c. ). 
It was of course necessary to employ such verifications, as a test of the correctness of 
the several divisions, before proceeding to collect them together, but the collection itself 
affords an exceedingly good ultimate verification. The following is an exemplification : 
the terms in y which involve the product BCDE are obtained by the collection of the 
corresponding terms in the ten divisions, as follows : 
123 456789 10 
BCDE . - 5 aY‘ 
+ 1 
- 2 
— 2 
+ 1 
+ 1 
- 2 
— 2 
+ 1 
- 2 
+ 1 
+ 30 abef 
+ 49 
- 16 
— 20 
+ 4 
— 25 
+ 50 
— 20 
- 25 
- 16 
+ 49 
+ 980 aedf 
+ 80 
+ 200 
+ 184 
— 148 
+ 100 
+ 184 
+ 100 
+ 200 
+ 80 
— 280 ac^ 
+ 80 
+ £0 
— 440 
— 180 ad^e 
- 60 
- 60 
- 60 
— 280 ¥df 
— 440 
, 
+ 80 
• 
• 
+ 80 
— 825 6V 
— 825 
- 180 bdf 
• 
- 60 
* 
- 60 
• 
- 60 
+ 740 bede 
+ 740 
bd? 
c^e 
<rd^ 
+ 1750=0 
+ 210 
+ 122 
+ 182 
— 1228 
+ 76 
+ 48 
+ 182 
+ 76 
+ 122 
+ 210 
= + 1228 
4 H 2 
