74 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
Expanding this in powers of y, it is easily proved by successive approximation that 
y 
or 
X<"> = (2^i+ 1)^ - 
“1“ VTj + -I K 
a oa IDa 
If) 
13 X 32 
{2n+\)Tr 3(2?i + l)V' 
= 1-57079 (2?^ + 1) — - 
•63662 
•I720I 
15 {2n + 1) V 
•03558 
2n+l (2n + If 
The first root is by numerical calculation 
X = 4-49341'= 257° 27' 10" 
The foregoing formula gives for this case [n =1) 
X = 4-49366. 
For higher values the formula is correct to five places at least 
The first three roots are 
4-49341 = 270° 
7-72528 = 450° 
10-90408 = 630° 
12° 32' 50 
7° 22' 27" 
5° 14' 23" J 
The Xo parameters are roots of the equation 
cot X = ^ 
A/ o 
The large roots are clearly nearly mr — mr — y say, where 
nir — y I 
cot y 
niT — y 
or 
cos y 
niT — y 
1 
?i-7r — y 
Sin y. 
■ (30)- 
(31). 
Writing and expanding in terms of y it is easy to prove, as in the former 
case, that 
3 
X = niT — 
= 3 - 14159 ^^ — 
3 
3 
./O'. 
^ (,V' 
mr \mr! 
95493 -29026 -15881 
n 
w 
(32). 
