1909-10.] Illuminating Power of Groups of Pin-hole Burners. 53 
thus two of the more important points in each graph are liable to an error 
which would make a considerable difference in the inclination of the straight 
line referred to on page 51. /3 is, of course, the tangent of the angle of 
inclination. 
For the purpose of comparing the results obtained in the various cases — 
two, three, and four burners — it will be desirable to examine turning points 
and inflexion points. Writing the equation for P in the form 
P=1 +ae~ bx2+cx , 
c 
then the turning point of any of the graphs in fig. 3 is given by x = -y . 
This value of x for series C, D, E is respectively +*83, —T9, and +*61. 
Thus D does not give a turning value for any distance of the flames, while 
C and E lead us to expect a turning value for a distance about ’7 cm. The 
burners could not approach so near as this. As in fig. 3, it is consistent 
with observation to suppose that the A and B curves show curvature only 
in one direction ; but it is also consistent to suppose that they are of the 
same general form as the other graphs, but with inflection points nearer 
the origin, and no turning values. Thus series A and B may be classed 
with series D. Now C, D, E have maximum increase in P (calculated) = 
35 per cent., 47 per cent., and 33 per cent, respectively. Since the A and B 
curves are like the D one, only with more uniform curvature, it may be 
assumed the maximum increase for A and B is something like 50 per cent, 
at least. Thus the maximum increase of P for the two burners is probably 
over 40 per cent, at least on an average. 
The inflexion point further to the right of the origin is given by 
35 = ^ 4 - This value of x for the C, D, E observations is I’ll, *67, 
and 1*04 cm. respectively. Taking the A and B observations into account, 
and assuming the inflexion points for these series of observations to occur 
at a distance not greater than '7 cm., we get an average distance not 
greater than ’85 cm. 
The simpler formula for P holds very well for the smaller values of x, 
which are the more important values. Series B, D, E have, roughly, the 
same formula : 
P = 1 + l”2e~ v8x J 
while A and C are satisfied by 
P --= 1 + e~ v4x . 
Thus a very rough formula to satisfy all the cases at small distances is 
P = l+e-P. 
