EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
381 
In interpreting this formula, it will be noted that, if i is less than Jc, the term 
vanishes : hence the summation runs from k = oo to k — i; it is therefore better to 
write k + i for k, and we thus obtain 
c* W; 1 k z2° k + il [a\ ,c Wk 
— _ v 
R 2i 
+ 1 
c k = o i\ k\ \c a 
( 4 ) 
This is the first transference formula by which the solid zonal harmonic of degree 
— i — 1 with respect to O is expressed as a series of solid harmonics of positive 
degree with respect to o. The formula (4) includes (2) as the particular case where 
i = 0. The right-hand side of (4) is convergent for r less than a. A similar formula, 
convergent for r greater than a, is easily obtainable, but with this we shall not 
concern ourselves. 
It remains to find the transference formula for certain tesseral harmonics. 
If we put 
p = i(^ 3 + r),.( 5 ) 
the general expression for the zonal harmonic is 
Wi = 
i\ 
k?.i- 2 k\ 
z l ~ 2 h p\ 
where the summation extends from k — 0 to k — \i or \{i —l ). 
From (6) we have 
rJ'tll- ■ ri ! 
: ___ Jc- 1 
k?.i-2kr z p • ' ’ • 
y/u_ 'JL 
dp ~' ' 
Now, since r 2 = z 2 + 4p, we have 
(-YH 
dp v ' 
— (Jc + 1 ) 
k + V?.i - 2k - 21 1 kl 2 .i-2kl 
+ 
4k 
zi — Zkpk 
_*/ u (* +!)-(*- 2*0 (* ~ 2 * ~ * 
Also 
2(2i+l)«* = S 
V- 
Subtracting (9) from (8), and simplifying the difference, we have 
j c +1 d + 1 ) (i + 2 ) . i\ ni _ u j. 
:\~.i - 
i + 2 ! 
Jp - 2 + *> * *= *(-)” 1 {k + l ) MKi-2M * ~ ' 
( 6 ) 
( 7 ) 
( 8 ) 
19) 
— v /_u+ i_ 
’ k + l\Ki + 2 - 2k - 2! 
d 
-{k + + 
dp 
IV 
M- 2 3 
( 10 ) 
