24 Prof. W. G. Adams on the Forms of 



If r = a, then p + cos 0=1. 



da 



Hence r . — - = ; and the curves generally cut the edge of the disk at 



right angles. 



"When fi=2 and cos = — 1, then the two branches of the curve meet 

 on the edge at the extremity of the diameter through the two poles. 



On this curve 



r d6 = V(2 + cose) 2 -l _ V(l + cos0)(3 + cos0) == / 3 + cos0 

 *dr s in sin V 1-cosfl' 



Hence when cos 0= — 1, 



eld ,, 

 r dr =±1 ' 



and the two branches of the curve are each inclined at an angle of 45° 

 to the edge of the disk. 



The polar equation to this curve is very much simplified when this 

 double point is taken as the origin of coordinates. 



The polar equation then becomes 



r 2 -f- 4a 2 — 4ar cos = 4a *J r 2 + a 2 — 2ar cos . r 4 — 8ar 3 cos 

 + 16aV cos 2 + 8aV = 16aV . (r - 4a cos 0) 2 

 = 8a 2 .r=4acos d±2»/2-a; 



and — = —4a sin 0. 

 dd 



From this result it is easy to see that there is no point of inflection in 

 the curve ; so that each branch of the curve is continuous and remains on 

 the same side of its tangent. 



From the above equations the curve may be traced. It consists of 

 two loops, which form a continuous curve, the two branches cutting one 

 another at right angles in passing through the double point on the edge 

 of the disk. 



The complete equip otential curves for the unlimited sheet may also be 

 readily traced from the above equations to those curves by giving different 

 values to /* (see Plate 1. fig. 7). 



Neglecting the part of the curve outside the disk, the equation to the 

 curve may be put under the form 



{r-a(, + e0S 6)} = _ V (M -l +co8fl)(/ , + 1+C o 8 0). 



CI 



To find the points where it cuts the axis, let cos 0= + 1 ; then 

 where the upper signs are to be taken together. 



