72 
MR. J. H. JEAXS OX THE EQUILIBRIUM 
given by equation (1). I have made no attempt to discuss these results from the 
point of view of pure mathematics; we therefore pass at once to some simple 
illustrations of our theory. 
Circulav CyIinde ?\ 
§ 6 . Let us attempt to find the potential, cross-section, and centre of gravity of the 
cylinder, of which the' cross-section is the circle 
~ c)' + .V~ = f'"'.(-3)> 
or, in polar co-ordinates, 
r- = cr -f lev cos 6 — c~ .( 2 - 1 ). 
The 7] equation to this curve is 
= ((-- + c + v) — c- .(25), 
or, solving explicitly for f, 
f = c + cd;{r) — c). 
Now the minimum value of | | which can occur on the curve is a — c, and this is 
greater than c if a > 2 c. In this case we have 
1 If c d \ 
= ^ + •••’ 
rj — a 7] \ 7] rj~ j 
and therefore the solution for * is 
n / n s on 
, fr , c , c- , \ , rr , ca~ , 
^ — c+~(lH-h~r+---) — c-f- -^ 
7 / \ 7j 7j- / 7) 7]-^ 
We accordingly have 
V. = 7rp[C+c(f-h7?)-|p] .(26). 
V, = I - cr log f, 4- (A -h I) -Fi c-V (A + U + . , , } , (27). 
A = ttcA Aa = Ticcd. 
If we determine C fi'om the condition that Vq and shall become equal at the 
boundary, we have the known values for Vq, V;. The equations for A and a. reduce 
to A = TTO", a. — c, and these, again, are obviously in agreement with the. 
known results. 
Ellijjtic Cylinder. 
§ 7. Let us next find the potential of the elliptic cylinder 
ax^ + hy- =1 .(^ 8 ). 
The 7} equation to the surface is 
{a — />) ( 4 -'' "h 77 ") -f- 2 {a -f- 1>) ^7] = 4, 
or, solving explicitly for 
I I ~ .^) d "1" + I (« • • • • (-^)- 
