[889 
8 8 9 J 
AND THE CONSTRUCTION OF MILNER’S LAMP. 
5 
e have 
id, 
0). 
the conclusion is that 
>f an arbitrary constant 
i 6, but not so that it 
e this value, we must 
is of a, /3, r 0 ), i.e. we 
,ting r 0 as a disposable 
here C=r<? — 2ar 0 cos /3. 
it with r 0 arbitrary, but 
> or r 0 = 2a cos ¡3, giving 
o be able to satisfy the 
s between the constants 
where the terms containing /3 are readily reduced to fa 3 cos 3 /3 sin (6 — (3) ; hence also 
the terms without /8 disappear of themselves : and we have 
a cos 6. X — | ( Y cos 6 + Z sin 6) = |a 3 cos 3 (3. sin (6 — /3), 
which may be put 
= b 3 cos (6 + a): 
viz. this will be so if we have the two relations 
a = ^TT — /3; and b 3 = — fa 3 cos 3 /3. 
I make (see figure) Milner’s lamp, with a circular section, /3 arbitrary, but a 
B 
segment AM (zSAM = ¡3) made solid. G in the line SG at right angles to AM 
is the c. G. of the lamp, and G' the c. G. of the oil. 
And this seems to be the only form—for the pole of r must, it seems to me, 
be on the bounding circle—viz. in the equation r 2 — 2ar cos 0 = C, we must have (7=0. 
4/3 — sin 40)} 
s 4 /3 — cos 4 0)}