OF EQUILIBPvIlT!\[ OF A ROTATING !\IASS OF LIQUID. 
BOo 
§ 2. Change of Notation. 
It will be convenient, with a view to future work, to change the notation, and I 
desire to adopt a notation which shall not only agree in the main with that used 
in “ Harmonics,” hut shall also facilitate reference to a previous paper on Jacobi’s 
ellipsoid (‘ Roy. Soc. Proc.,’ vol. 41, pp. 319-336). 
I write 
1-^ 
1 + ^’ 
f'l 
K ~ 
o 
K''. 
It may lie noted that what I liere write k was denoted hy k in “ Harmonics,” and 
vice versa. 
I have in general written the current co-ordinates v, p, f/), and the ellijisoid of 
reference r,,, so that the s(piares of the semi-axes are 
/.:■ 
1- + /3\ 
f - /3 / 
1 ), 
I now propose to write as llie s(piares of three semi-axes of 1lie ellipsoid of 
reference 
O O 0/-1 o*o\ o 
C-COS"y, C'(l —K'Sury), C". 
Comparing these two we see that 
/,- = CK sin y, and Vr. = —:—. 
' ’ /C Rill 7 
For the current co-ordinates I retain (f) and write 
1 
= sin 0. 
K sm ip 
The three roots of the fundamental ciiliic are therefore 
V 
o • o I ) 
AT" snr Y 
0 • 0 /I 
fX- = SllV U, 
^ cos 2(^ 1 , ^ i\ 
= . 0(1 -x-cosw/)). 
1-/3 
Tlie rectangular co-ordinates x, y, z are therefore now expressible as follows : — 
X = — —, , cos Q ( 1 — sill'' d)- cos (b, 
sill yjr ^ / r’ 
. (1 - /C~ siiF i/i)’ COS e sill <f), > . ( 8 ). 
. = sin^(l-.'^cos^</>)i 
sin ip ' T / j 
2 B 
VOL. cxcvni. -A. 
