48 
Proceedings of the Royal Society of Edinburgh. [S 
ess.* 
It can be shown * that R = B0 + /q where h is a small quantity with the 
values given on following page. 
* Value of h. Let </>B = r, d = angle I<pB , x = distance of adjacent burners, 
a = distance of the burner referred to from the centre of the cross-piece (B), rj_ = 0l, 
r 3 = 03, r 4 = <£4. 
(1) Case of two burners. Here R = r 1? therefore 
B = 0l =r sec 0=r+r— =r+ — 
r 2 8 r 
(2) Case of three burners : 
srl(;? + i7«) bydefinition - 
As above, 
Also 
r,=r + — . 
1 8 r 
, V3 
r 3 = r+^-x 
Instead of - — E_ write E _ ?£ + EL where e is small. 
(■ r + e ) 2 r 2 r 3 r 4 
1 
R 2 
1 
o+: 
7 x 2 
r 2 V3 r 3 ' 12 r* 
= ^+* sa y> 
giving the addition that must be made to E . To find the corresponding addition to 
r , let E =y. 
r=—E . Let h be the addition to r, 
\ /y 
_ l 1_ h_ 1 3 k 2 
Vfo+*) " *Jy~ % y : * + 8 yi 
= r 
v -f- h ■ 
— r 3 + -k 2 r* 
2 8 
(3) Case of four burners : 
x x z 
2*y3 6r 
1 _1 f 2 
R 2_ 4 
IWA}- 
trp r 3 2 r 4 2 J 
Now 
1 = r + "8T =r+ 2r 
r 3 =r + a 
r A = r- a . 
As before, write 
1 2e , 3e 2 . 
+ — r — etc. 
(r + e) 2 r 2 r 3 ?- 4 
1 J 1 f a 2 23 aU 
" R2- r 2 + 4 \ 4 r 4 + 8 r° j 
Neglecting the last term, which is very small, we have 
But if R = r + /t, we have 
p_l a 2 
R 2 “r 2 + r 4 
112 h . 
U 2== ?~ ^ approximately. 
h= — — = — 
2r 
4r 
