1888-89.] Dr Sang on Air's Resistance to Oscillating Bodies. 185 
( 2 n+i)t v = — o. 2n+2 x + (4B + 3)a 2n /3v 2 — (4B + 2)a 2n+2 ftx 2 
+ (10B + 7C + 6D + 8)a 2n /3 2 xv 2 - (2C + D)a 2n+2 p 2 x B ; 
and by help of this scheme we form the following table of the 
numerical coefficients : — 
X 
fiv 2 
jto 2 
fi 2 XV 2 
jS 2 ^ 
In - 1 
+ 1 
— 
A 
+ 
B 
— 
C 
+ 
D 
2?i + l 
-1 
+ (4B + 3) 
- 
(4B + 2) 
+ (10B + 7C + 6D + 8) 
- 
(2C + 3D) 
1 
-1 
+ 
1 
0 
0 
0 
3 
+ 1 
- 
3 
+ 
2 
- 
8 
0 
5 
-1 
+ 
11 
- 
10 
+ 
84 
- 
16 
7 
+ 1 
- 
43 
+ 
42 
- 
792 
+ 
216 
9 
-1 
+ 
171 
- 
170 
+ 
7268 
- 
2232 
11 
+ 1 
- 
683 
+ 
682 
- 
65976 
+ 
21232 
13 
-1 
+ 
2731 
- 
2730 
696052 
- 
195648 
&c. 
&c. 
&c. 
&c. 
&c. 
&c. 
Whence 
v ) at aH B aH b aJtf ) 
= "' tX I r-iX3 + i^-T^ +&c -/ 
+ apx* | -16^ + 216~-2232^ + & c . } . 
The series by which -aX is multiplied is that for sinctf. The 
series of coefficients B may be written 
2 = 2{l}j 10 = 2{l + 4}; 42 = 2(1 + 4 + 16}, and so on; 
or in the more concise form, 
2 = | (2 2 - 1) ; 10 = ^(2 4 -1); 42 = J (2 6 - 1), and so on ; 
wherefore the term involving /3 may be written 
1 OV2 / 2W ^ 
3 a/3X \ 1727 
3 1...5 +&C - - 2 17^3 + 2 
a 5 tP 
...0 
&C. I 
or supplying the defective terms, by adding zero in the form 
