FIGUEE OF EQUILIBEIUM OF A ROTATING MASS OF EIQUID. 
291 
+ Sa'yS'Gs.i + (2ay + (2a' 8 + 2^ y ) Oo^ 
+ (2^' S' + y^) GL + 2y S'ni° + 8'^n’y 
From the definitions of o-^., in ( 8 ), we see that 
= yyf.C‘fi -[/(-0/(no) -,/'(A.)/(a,.)] + e[/(A,)/(o„) -/(Ao)/(o,)] 
TT bill y 
- /^[/(As)/(n,) - ./■(Ar)/(n,)J], 
A 00, 3 sin ^ (f [/(A,)/(n„) -/(a,0/(A,)] 
- 3r;[,/’(A,)/(n,) -/(Ay/(n,)]}. 
O’-!. = 
The computations (which were in this case actually made from the corresponding 
formulae involving the IT, T integrals) gave 
0 -, = •0136866, Cr = ’00009246, o-^ = -00176135. 
These have to be used in a formula which also involves '^ 3 . Now ^3 denotes 
^ 13303 , or what should he the same thing, Pi^Qi^. The formulje in the “ Pear-shaped 
Figure” with y = 69° 49''0, k = sin 73° 54'’2, give 
== -351697, = -351744. 
Thus the two functions, which should l)e identical in value, differ by -000047. I 
think that if I had taken y = 69° 48'-997, k = sin 73° 54'-225 (the actual numerical 
solution for the critical Jacobian, although not fully stated in the “ Pear-shaped 
Figure ”) this small discrepancy would have been removed. However, the difference 
is quite unimportant, and as generally means Pi^Qi^, I take the former value and 
put log "^3 = 9-54617. 
With this value I find the required result, namely 
Ha [i (o-,f + 2y - i<r,, = - -00050012 .(43). 
§ 17. The Integrals oj/, p/, (^f. 
Any harmonic *S'A where i and s are Ijoth even, whether in the approximate form 
of “ Flarmonics ” or in the rigorous form, may be written 
Nf = [a — h COS” d -f c cos^ 6 ~ d cos® i9 -f- .. .) (cd + 6 ' siiP </> + c' sin^ ff) d' sin® 
Each series is, of course, terminable, the number of terms in each of the two factors 
being -\- 1 . 
For the determination of the &>, p integrals this must be multiplied by (bg)^. It 
2 i> 2 
