EQUILIBRIUM OP ROTATING MASSES OP FLUID. 
38i 
Hence 
[k h y] = ~ E ' Ei ■ y l °g ( 1 + P)> \J C > a r] = E k E l . r log (1 + i?). ( 29 ) 
We must now develop the symbolical sums of the series in (28) and (29). 
The following theorems are obvious :— 
d n 
dy 
nT 
•P — 
p — n ! 
7 
,p — n 
d n , n m. n — 1! 
(1 _ 7 )» 
d n . v + n — 1 ! , \ 
-^(1 _ u (1 y )~ ?| —'*■ 
Then, by their aid, we have from Leibnitz’s theorem— 
|)/mog(i +/3 )=;r^ ![V 
t = Jc 
t = o ^ ^ ! 
fit fjk-t 
/ + 1 J^og(l+/3), 
klk + 1! k - t - 1! 7*-* + 1 
t + 1 ! (1 - 7 )* 
-< 
in which we interpret (— 1)!/(1 — y)° as log (1 + (3). 
Thus 
/ 7, \ __ % k _^_ k + 1\ 
' ’ V' ~~ t to (k - t + 1) (k - t) t ! k - t ! 
with (3 °/0 = log(l + /3). 
Again 
^|Ur)(U’iog(i + /3)) 
_ 1_ * = * * + Ilk- t - 1! d ; 7 J + *- < + 1 
- . . 
P 
<lc-t 
Ic-t 
(30) 
Hence 
[£, i, y] 
Z = & r = f 
= 22 
i \ t = Q t \ k — t \ k — t + 1 ! dy 1 (1 — 7 ) 
* = * ■'■ = i £+1!7 l — £ — l!fr + fc — £ + 1 — 1! 
y t = 0 r = 0 t \ k — t \ r \ i — r \ k — t + 1 1 k — t — ] ! i + k — r — t + 1! 
0 
i + k— r — t 
= 0 r = 0 + k — ?’ — 7 + 1) (i + k 
k T~ 1 ! % -f- /,; — 7 1! 
7) & — t -P I! 
P 
i + Jc — r— l 
t\ k — t\ r! i — r! 
(31) 
In (30) and (31) the infinite series are replaced by finite series. 
From the form of the series it is obvious that the result must be symmetrical with 
respect to k and i, so that [k, i, y] = \i, k, y], but this is not obvious on the face of 
the formula (31). 
We shall find, therefore, the symmetrical form of (31) for the first few terms. 
If t = k, v — i, we obviously have 
First term = (k + 1) (i -f- 1 ) log (1 + P)- 
The second term arises from t — k, r — i —1, and t — k — 1, r — i. The two 
corresponding values of (31) will be found to add together, and we get 
Second term — ^ {k + 1) (i + 1) [2 (i -f k) + ik] (3. 
3 d 2 
