370 
TABLES OF THE DEVELOPMENTS OF FUNCTIONS IN THE 
[216 
1 n cos 
2 sin 
C P 
[ P 
[ P 
[J 
e 4 e G 
e 6 
e 7 
768 92160 
7680 1290240 
92160 
1290240 
[cos] 
+ 299 - 74352 
+ 3760 -1457120 
+ 57262 
+1025248 
[sin] 
+ 326 - 75648 
4-4016 —1482208 
+ 60387 
+1071904 
[cos + sin] 
4- 625 — 150000 
+ 7776 -2939328 
+ 117649 
4- 2097152 
[cos — sin] 
— 27 4- 1296 
- 256 4- 25088 
- 3 I2 5 
- 46656 
where the lines [cos + sin], [cos —sin] are obtained by the addition and subtraction of 
the lines [cos], [sin]. 
The lines [cos + sin], [cos — sin], are then rearranged by reading them backwards 
and forwards from the terms which are in the [cos — sin] line, under [ ]*’; viz., in 
the present case, where j = 1, from the terms — 2, 0, 0, 0 under [ ] J , as follows:— 
[ ]° 
CP 
x° C ° S f 
snr 
e 
e 3 
e 5 
¿> 
e° 
e 2 
e 4 
e 6 
4 
96 
7680 
1290240 
2 
16 
768 
92160 
[cos + sin] 
— 2 
0 
0 
0 
+ 1 
- 8 
+ 42 
— 800 
i [cos — sin] 
Do. 
99 
>9 
99 
0 
— 1 
+ 8 
+ 555 
[cos] 
- 4 
0 
0 
0 
+ 1 
-9 
+ 5° 
- 245 
[sin] 
0 
0 
0 
0 
+ 1 
- 7 
+ 34 
- 1355 
,° C ° S f 
sin' 
[ p 
[ P 
[ P 
e e 3 e 5 e 7 
4 96 7680 1290240 
e 2 e 4 e 6 
16 768 92160 
e 3 e 5 e 7 
96 7680 1290240 
l : [cos + sin] 
4-2—60 4-1360 —31360 
4-9 -648 4-34425 
4-64 —8960 4-892416 
\ [cos - sin] 
0—44- 80 4- 2688 
O — 27 4- 1296 
0 — 256 4- 25088 
[cos] 
4-2—64 4-1440 —28672 
+ 9 - 6 75 + 35721 
4-64 —9216 4-917504 
[sin] 
4-2—56 4-1280 —24048 
4- 9 —621 4- 33129 
4-64 —8704 4-867328