214 Mr. Atry on the Figure of a Fluid Mass 
(22.) Since C is indeterminate, it is manifest that there is 
nothing to determine the value of the constant term in x (c) 
and it is, therefore, quite arbitrary: It may conveniently be 
taken so as to make x(c) vanish when c=r. This gives 
x (c)= — Q (r?—c°)— R (r4#—c*) — S (7° —c®) — T (78 — 8) = — (7? —€’) 
x {Q+ Rr 48r+ Tr +R+ Sr? 4+ Tri.c+ 8 + Tr?.ct + Tc} 
AVA ce 310509 Br? ox oupa 6 ert 99. Dy 5 bs iB 
pre ex} 11.83.14.25. 000 267.1010 Gk Was rk 4s, 
11013 .14,95 8+) los 
1443 Fre 39 E 51 F 
14.25° k *10°% 14k“ 
(23.) As an instance of the application of this formula, let 
it be required to find the figure of Saturn, as affected by his 
rotation about his axis, and by the attraction of his ring. Let T be 
the time of his diurnal revolution: the resolved parts of the cen- 
trifugal force on a point whose co-ordinates in the plane of his 
27\° 
equator are a, b, will be Te @ and ag b. The part of U which 
236589 Fr 519) Er’ \'9 DY (2 
2°k 
arises from this, will hee 7 = ae + 6) = asd ar —c*), and, therefore, 
the part of «(c) will be = (r?—c), or — are neglecting the con- 
stant term. 
(24.) Suppose Saturn’s ring to be a mathematical line, into 
which is collected a quantity of matter = -th of the matter in 
23 
the body of Saturn = 2a: let its radius= R. The part of U or of V 
which arises from this is 
ark f° 1 bar he 1 
3n YOV{R+a*+2aRcos.0+c%} — 3n/40Y{R+7r°4+2aR cos. 0} 
