OF EQITILIBEIU.AI OF A ROTATING MASS OF LIQ,UII). 
311 
where 
so that 
From (16) 
f/ = T (1 4 - 2k- =F (1 - -f iK^f ), and /c" = 
2 - 
r?' 
K" — 
- = 1 - 4r 
, 0 M3 
ru 
W {-) W il^) ^3^ {i>) and w (-) W (^) ^ 3 ^ {i>) = V ir^ + ?/ - 
+ G-q^q^' sin- y) 
o ^ “ GQ" 
Avhere 
so that 
Lastly, from (IT), 
i (2 -h /c~ 4 - (d — K'K-)-) and /c'" = q 
\ - 2 ^^ 
(23), 
P.ROP/iriS/(.*) = «/:. 
It is easy to verify that each of these expressions satisfies Laplace’s e(|uation 
(24). 
^ 4. The Exqwession for the Q-fimctions in Elliptic Integrals. 
In this pajDer 1 drop the factors (!H and E whicli Avere found to be necessary when 
the Q-functions were expressed in series. 
We make the following definition :— 
(y,) = H9/(G))]~ [ 
ilv 
1 + ^ y, ) 
1 - y 
and a similar formnla holds for P/ Q;. 
It is clear that p/ may he multiplied by any constant factor without changing- 
tire resnlt ; hence we may use the forms which have been found in §§ 2, 3, 
The notation must uoav he changed. 
We have 1 /= —r—r and ca =—^r -^. Therefore, when rjj is adopted as varialile, 
K sin ->|r K sin 7 ^ i. ? 
the limits are y to 0 , and the sign of the whole is changed. 
But 
ch = 
cos y{r 
K sin- p 
dxfj, 
and 
1 + /Sf _ C0S'v/r(l — /c-sild-v/r)' . 
1 — /3/ /c-foin-y- 
dv 
^ 1 )^ (M- 
iJL^A — 
I - y 
d-p 
(1 - K-sh-r-pf- ' 
Therefore 
