100 
ME. J. H. JEANS ON THE EQUILIBEIUM 
be good, the root being IMS, and the error occurring only in tlie tbird decimal jjlace. 
As <f) decreases the approximation improves, and at <f) = TTj'l the error occui'S only 
in the fifth decimal place. At (f) = 7 the error again appears in the third place, 
and after this the approximation is bad. My plan vas to calculate for smaller values 
of (j) as well as I could, taking care to keep the values of r in defect rather than 
excess of their true 's^alues. The curve was then plotted out on paper ruled with 
squares of 1 millim., the unit of length being taken to be 50 millims.'^' 
The area of the elliptic cylinder of tig. 4 is known to be 
13,000 sq. millims., 
and this would also have been the area of the present curve had it been accurately 
drawn. The area of the curve (obtained by counting squares) was, however, found 
to be 
12,776 sq. millims. 
The moments about the axis <p = 7 t,‘1 of the two parts of the curve {<f) > 7t/2 and 
(^<7r;'2) ought of course to be equal : these were found to be respectively 
208,290 cub. millims. and 302,850 cub. millims. 
It was therefore obvious that the curve had been too much shortened in the region 
in which ^ < 7T°. 
Keadjusting the curve in this region so as to divide the error as ee[uaUy as possible, 
I arrived at a cui've Avith outstanding errors in area and moment of 
130 sq. millims. and — 130 sq. millims. at r = 125 millims., (f) = 0. 
This is the curve given in hg. 5. It Avill be seen that the error is one of about 
1 per cent. 
Calculation of the Curve 9—■ i. (Fig. G.) 
§ 31. The equation of the curve is found to be 
r-= 1-162— -2117'CoS 0+ •503;-'cos 2^-f -053?’^ cos 3(^ + -Ollr^cosI^ 
+ 'OOISF cos 50 + -UUl Ir*" cos 60 + ’GOOISr" cos 70 
4* [■00U2r^ cos 80 + 'OOUI r'Aos 00 + . . .j 
the terms in square brackets being those mentioned at the end of § 20. 
* This lius been photographically reduced to 25 millims. before printing. 
