On Photometric Paddle-Wheels. 
675 
1910-11.] 
In the above discussion only wheels of the A and B types have been 
considered, but before passing on to the others it is first of all necessary to 
treat the theory somewhat more generally than has yet been done. If t 0 
and e be roughly plotted as functions of a, the graph shown in fig. 12 will 
be obtained, e being here defined as the change in e 0 caused by altering X 
from zero to some fixed maximum value. Now, for the medial rays of 
the beam X = 0, and from (10) and (11), 
1 -* 0 = »E 0 /360 = ?i(K + | e 0 | )/360 ; 
while for the most lateral rays, X is a maximum, e 0 becomes (e 0 — e), and 
(from the above equation) t 0 becomes (t 0 + ne/360). Hence the change in t 0 
is simply proportional to e. But unfortunately it appears from fig. 12 
that when t 0 is least, e, and therefore the change in t 0 , is greatest ; i.e. things 
so fall out that the greatest absolute variation occurs in the value of t 0 
precisely when the value of t 0 is such that it will be most disturbed 
by any change. 
This state of affairs reaffirms a conclusion already arrived at, namely, 
that X' attains its minimum value X m when a is a maximum ; and it also 
accounts for the fact — which a little experimenting with the charts will 
soon elicit — that the use of paddle-wheels of the A and B types necessitates 
a comparatively severe restriction on the width of the beam. 
Much better results, however, can be obtained from the A' and B' 
types of wheel. In equations (A 7 ) and (B') the sign of cr depends only 
on the way in which the positive and negative directions are chosen for 
a, and, as this choice is quite arbitrary, cr can always be made positive. 
We shall assume that this has been done; and further that cr — to avoid 
repetition of values of E — is taken not less than the largest numerical 
value of e 0 . With this understanding (A') and (B') may be written as 
one equation, 
E = K + e (17) 
where K is a positive constant always greater than e. 
Hence from (10) and the above equation, when X = o, 
1 -t 0 = nE 0 /36O = n(K + e o )/360 .... (18) 
and when X is a maximum, e 0 becomes (e 0 — e), and from (18), t 0 becomes 
(t 0 + ne/360) ; so that the change in t 0 is proportional to e, as before. But 
now t 0 is a different function of a, and on reference to fig. 13 (which like 
fig. 1 2 is only diagrammatic) it will be seen that by restricting a to a range 
of negative values terminating at the origin, we have the power to so 
