50 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
In which it is clear that 
h=i 
h-. 
If the equation is 
{x + 1) — ^{x- + X 1), 
the positive root of which is x = 1. In this case 
h,) + i — I ^ = i. 
If ever is nearly F = J- -f- a (say), x^, is nearly I = ] -|- | (say). Then, regarding 
a and £ of same order 
and 
^j)+i — I 
(1+3^+ 3f) (2 + 0 = I (3 + 3^ + a (1 + 2a) 
^ = 5 « + 5 — If' = 5 “ — f t = 5 “ ( 1 — 1-5 «) 
i + « 
= 1 - (4 + a) (1 — 4a H- ^5^ a-) = i + a - f f ah 
Hence continually converges to ^ and the value of to 1 asj; increases. 
The first seven values are 
T, = 1-3283, 
6, = 
-7582. 
T., = 1-1840, 
lh = 
-6741. 
X; — 1-1284, 
h = 
-6315. 
a;4 = 1-0987, 
h = 
-6056. 
Ts = 1-0802, 
b.= 
-5882. 
a-g = 1-0674, 
h,= 
-5753. 
x-i — 1-0580, 
cx 
11 
-5660. 
The succeeding values will be given to four figures by the foregoing approximations. 
Tlie velocities of translation of the series of aggregates are 
Monad 
V, = 
1 - 5 - 
Dyad 
V2 = 
i(7- 
5xf) Vi 
Triad 
V3 = 
IOTi 4- 5Tia’2) Vi 
4-ad 
V4 = 
1(7- 
lOxi 10 X 1 X 2 — 
5-ad 
V 5 = 
i{7- 
lOxi + lOXix'i — 
&c. 
Vg 
V, 
V« 
= V, 
= — -9110 V,. 
= + -8615 Vj. 
bx\xlx\)^y — — '8282 ^^ 1 . 
lOxi.roXa + bx\xlxlx‘^^ V, = *8023 V,. 
= - -7833 V,. 
= -7618 Vj. 
= - 7462 Vj. 
