FIGURE OF EQUILIBPJUM OF A ROTATING MASS OF LIQUID. 
289 
a = 
C'lo = 
1.3.5 
2.4.6 
1 .. . 7 
2 .. . 8 ^ 
64 
1 + P + 1/5® + f/5® + (1 + /5 + ip~ ■” ^P^^ 
^ P T/5® + ^P^ + ~^5P‘^ + .7]^ (1 + /5 + f/5® ~ Jp^ iP^) 
128 
+ (1 + /5 + 0 X p® — 2/3^ -f 
O Oc^ 
C — 
^12 — • 
Ca - 
Ca — 
1 - + i-x®, c, = 1 + X - 2X'2 + IX^ c, 1 4- X + 0 X X2 - 2X'-^ + X4 
1 4 - X + |X2 - |X3 „ + X4 Cg = 1 4- X 4 fX2 - fX3 - I’X^ - 4X'^ + fx^. 
By means of these formulae I then formed a table of the IT, T' integrals, 
corresponding to the critical Jacobian for which y = 69°49''0, k — sin 78°54'’2. 
A little consideration will show that if nS,,, n|„ ITl,j . . . are a series of IT Integrals, 
the A integrals as defined in (34) are as follows :— 
AS„ = Ul, Al,_ = - AH 
2/1 
A 
4 _ 
A®nS 
A® — 
&c. 
Hence by differencing the IT integrals we find tlie A integrals, and similarly the 
differences of the T integrals give the H integrals. 
The converse is also true, and by differencing A, fl we return to IT, T. 
In this way I obtain a series of values of the required integrals. It may be tliat 
the last decimal place is erroneous In some cases, but the results given in the following 
table are sufficiently accurate for our purpose. 
Table of Logarithms of A and fl Integrals. 
n. 
log A!i,. 
log Ao«. 
log A.)". 
log 
log 
0 
•1064412 
•4317642 
•9049643 
1-4848513 
2-1218296 
2 
9-8518424 
9-9779679 
•1696589 
•4573383 
•8543000 
4 
9-7422028 
9-8168606 
9-9187709 
•0641097 
•2747811 
6 
9-6704204 
9-7230801 
9-7896484 
9-8771928 
9-9969099 
8 
9-6167543 
9-6573681 
9-7062560 
9-7667474 
9-8443343 
10 
9-5738269 
9-6068642 
9-6454046 
9-6909753 
9-7467821 
12 
9-5380322 
9-5658708 
9-59774.35 
9-6339429 
9-6770135 
n. 
log ll'ig. 
log 
log 
log 
log fig” 
0 
9-3901374 
•2047610 
1-0302812 
1-86670.36 
2-7138342 
2 
9-0333614 
9-8993680 
•7715371 
1-6492556 
2-5319082 
4 
8-888.3413 
9-7729763 
•6613027 
1-5528790 
2-4473548 
6 
8-7988101 
9-6930495 
•5898109 
1-4886133 
2-3893878 
8 
8-7344878 
9-6345770 
•5366725 
1-4398972 
2-3245748 
10 
8-6844351 
9-5884.555 
•4944045 
1-4005018 
2-2861489 
12 
8-6435159 
9-5503544 
•4594781 
1-3669609 
2-2533269 
2 p 
VOL. cc, —A. 
