bid 
207 
9, Let us now first consider the COSC» ES EF 6, — 0. 
Equation (23) becomes 
dp, ME eR uee yl ote’) 
SO 
p= ———__—. . .. . «,.s (29) 
Ve Re ] 
r 3 
The value 3a of 7, corresponding to z == 0, is, as is seen from (27), 
a value which is not attained. (29) shows that the motion takes place 
in a spiral which, extending to circle 7 =a, making there with the radius 
a finite angle, and, turning an infinite number of times, approaches 
to circle #7 == 8e on the inside. The particle can never get out of 
sphere r—3a and a motion such that the particle were from the 
beginning outside sphere =e (and such that e, 
—e,—e,=0), is 
: 4 dz \° 
impossible according to (28), as A should then be negative. 
ag 
] 
When 7 approaches to 3e then g approaches to — and conse- 
day 6 
| ; . 
quently the velocity to —. 
v VA) 
10. We now come to the case of two e’s being equal and differ- 
ent from the third. Calling (the three e’s being real) the greatest e, 
the smallest e,, we have two cases, viz. 
Ca = ee Ti Kle 
We first turn. to: ihe case ¢, = ¢, = —+4¢,. 
Excluding as before the interior of sphere 7 =a, 7 must be > «, 
so z< %. We put e, —e, = — a’, ¢, = 2a’; a be positive. Then 
(23) passes into 
— dz 
UP —-- 
(z + a?) V z—2a? 
It is seen that ¢ must be greater than 2a’, and, as z must be 
smaller than $, we must have 
Ph he ae Ne Wi IT || 
If 24° = 2, the particle is at rest on sphere =a. 
Now putting z= 2a’-+ 7’ we get 
— dy 
3 ap == 
“ y? + 3a’? 
and so 
