ME. E. A. SAMPSON: A NEW TEEATMENT OF OPTICAL ABEEEATIONS. 177 
The best way of forming the square, e.g., of 1)2, is to set up ‘99752 on the machine, 
multiply it into itself and into twice 392, then set up 392 and multiply it into itself 
and twice ‘99752, when the agreement of the middle terms is a check upon the 
operations. The last column, “ sum of coefficients in w,.,” will be used below as a 
check for future work. If necessary, it may be checked by the equation, e.g., for 0 ) 2 , 
-2‘38174 = -2‘37488x (‘99752 +‘00392)1 
Next, for formation of yjy, arrange as below :— 
f^r = 
r. 
Coefficient 
Coefficient 
Coefficient 
Coefficient 
Coefficient Coefficient 
Coefficient 
Coefficient 
Sum of CO- 
bo. 
^0- 
bo^. 
bol^o- 
bo‘^. 
bol^o- 
efficients. 
0 
* 
+ 1-00000 
1-00000 
-17045 
- -53998 
- -57232 
- -94185 
-32698 
2 
- -41285 
+ -65397 
+ -17045 
- -53998 
+ -42768 
4-72555 
-3-85792 
+ -55994 
+ 1-42757 
-76457 
4 
-2-21269 
+ -99379 
+ 4-89600 
-4-39790 
+ -98762 
-4-85284 
+ 4-14412 
- -61457 
- 1-32329 
-30504 
6 
- -20775 
+ -61078 
+ -04316 
- -25378 
+ -37305 
+ -73798 
- 1-50974 
+ -62230 
- -14946 
-81956 
8 
- -88382 
+ -99767 
+ -78114 
- 1-76352 
+ -99535 
+ -78114 
- 1-76352 
- -00465 
- -98703 
— 
The last row under ^8 '/—is the sum of the numbers above it and is introduced 
as a check upon the subtractions; it is equal to I3'q—/3q^. The addition of mixed 
positive and negative numbers is best done with a machine. We now have ■ 
r« 
r. 
Coefficient 
Coefficient 
Coefficient 
Sum of 
V. 
bopo- 
/3o^. 
coefficients. 
0 
+ -05573 
- -17656 
- -18714 
- -30797 
2 
+ 3-61299 
- 2-94963 
+-42811 
+ 1-09147 
4 
- 1 - 48032 
+ 1-26413 
- -18747 
- -40366 
6 
+ -60482 
- 1-23732 
+ -51001 
- -12249 
The multiplication by should be checked by the help of the column “ sum of 
coefficients.” It will be remembered that there is no check against setting up an 
erroneous multiplier for ^n,.. 
We are now ready to form (5G, &c. Referring to (12) on p. 157 and the calculations 
above we have, for example, 
^G= +‘98910a)o+l‘14882«2+lT7160«4+l‘24028««+‘01161V-o-’00830V-2-02057V^4 + *, 
the coefficient of \jr 2 , for example, being the figure that stands in the place ot G in 
the scheme formed on p. 175 for \l/- 2 ; and similarly for <511, (5K, ^L. It is unnecessary 
to write them at length because they are shown in a more convenient place in the 
following table:— 
VOL. ccxii. —A, 2 A 
