664 Proceedings of the Royal Society of Edinburgh. [Sess. 
two values for 0, i.e. determines two planes (7), the angle e between them 
being given by the equation, 
cos e 
1 + X 2 sec 2 a - 2c 2 tan 2 a 
1 + X 2 sec 2 a 
X being written for the ratio p/q, which may be called the position-ratio of 
the ray of light. 
2 sin 2 - = 1 - cos e 
tan a 
1 + X 2 sec 2 a 
. e c tan a 
girq 
2 + X 2 sec 2 a) 
( 8 ) 
The presence of X in this equation shows that the reduction of brightness 
caused by the vanes is different for different rays of the beam. It can only 
be sensibly the same for all the rays if the term involving X 2 is in all cases 
negligibly small in comparison with unity. When this is the case, then each 
ray undergoes sensibly the same reduction of brightness as those rays of the 
beam which are vertically above 0, fig. 1, and for all of which X = 0. The 
question of the greatest value that can be allowed to X (and therefore the 
greatest permissible width of the beam of light that can be used), with- 
out violating the condition that the term of (8) in which it appears is 
negligible, will be discussed later ; meantime we shall simply assume that 
the condition is complied with. On such an understanding (8) becomes 
sin = c tan a . . . . . (9) 
In order to facilitate the discovery of the properties and behaviour of 
any proposed design of paddle-wheel, the above equation, and all the sub- 
sequent equations of fundamental importance, have been “ graphically 
tabulated By this means the quantitative results for any paddle-wheel 
may be found at once without having to resort to calculation. 
The principle on which Chart I. is constructed will be understood by 
reference to fig. 9. OA, the scale along which c is measured, is of unit 
length, and is linearly divided. The arc AG, along which the angles 
are numbered off, has its centre at 0. If then AOB be the value of a, 
tan a = AB ; and if OD be the value of c, c tan a = CD = EF. But 
EF = sin EOF, therefore EOF must be the required value of ej 2, corre- 
sponding to the given values of a and of c. The actual chart has been 
drawn in two parts, (a) and (6), for the sake of greater accuracy, and there 
are some slight modifications in detail ; but these ought to present no 
difficulty: e.g. on the ~ scale the numbers have all been doubled, so that it 
gives not ^ but e Q , which of course is more convenient. 
To use either part 
