88 
ME. J. H. JEAXS OX THE EQUILIBRIUM 
Since satislies equation (82), ^^■e have 
^0 = 4^ ± i \/\hf — . 
or, expanding in the appropriate form, and taking a = 1, 
. 1,5, 50 , 1000 , 25,000 
&o — 2P + o + 5 ^ + n—• • • 
rtrj J / ?; •> 77^ o'?;' 
From equations (82) and (80) we have 
t - 3 n _ 1 3 j_ 5 I 1-5 2500 
U — — V — 2 — 4 ^ + 3 + 2^ + + • • • 
By a similar process we obtain 
fo-’ = - i) - !>)* - = W + i*) + s; + hh + . . 
Equation (87) can now be put into the foiiii 
5 25 
!>? 
6 ?; 27r)^ 
500 
O 
i)r) 8 It;’’ 
o ’I ^-1 '^^-3 
3Con~ "F C‘i — — — . . . 
Vi V 
■ ( 88 ), 
(89). 
■ (90)- 
• (91)- 
- Co 
50 . 1375 
9t; 81?;’ 
+ . . . 
+ Cl + 3 “;, + 27P + ■ ”J 
• (92)- 
Equating the coefficients of the various powers of p we obtain 
9 „ — 9^ 
gCg - gCg 
“k i<^l — 4^3 “t“ 2^‘l 
“ f^-1 — “^^3 ~F 
_ RA/-> _1_ ii/i 9/1 — 1 3 7 5 /1 I Ail/^ 
8 1^3 2 7'’! n 6^-1 8'--3 — 81 ^3 1 27^1- 
The lirst equation is, as it ought to be, an identity. We may assign to Cg any 
value, and therefore take iq = 1, this being equivalent to fixing the linear scale of 
measurement of d. Solving the remaining equations in succession, we obtain the 
following scheme of values :— 
F* - 1 F* 7!^ — 0 /• - - - X O 0 0 
^3 — -*-5 ^1 — 35 ^-1 — ^-3 — 81 • 
The vanishing of c_^ shows that the centre of gravity of the curve is, as it ought to 
be, at the origin. We now liave as the value of equation (8G), 
- -3" + 27T 
(93). 
§ 22. We now proceed to the determination of fo. Equation (84) takes the form 
iv - Uo) (U2 + |S2fo) = + §0^1 ( 3 ^ 0 ^ ^ Y) 
+ -Xu + ^ 2 ^,, (4” -k v") .(y-i)- 
/t=i 
The value of £0 is of the form (equation (78)), 
( 95 ). 
