92.6 REPORTS ON THE STATE OF SCIENCE, ETC. 
can be readily found by marking out the scale of @=0 to o in degrees on the extreme 
left of the abac: we shall call this the scale of 5, so that in general 
6=i0 + 8 
where # is an integer. 
If the initial argument (a) is expressed in the form to + 6, then a horizontal line 
drawn through 6 on the 6-scale will pass through the angles a—io,...a,a+0,.. 
and readings will commence on the line corresponding toto. In practice this 
horizontal line is lightly ruled in pencil, and is afterwards erased. 
There is, however, a limit to the number of readings on one horizontal line. We 
shall suppose that the abac covers 7 x 360° and contains v complete vertical sections 
and one incomplete section; then the length of the incomplete section in degrees 
will be 
vy = 3607 —vo. 
But if the abac were extended we should be getting precisely the same values as 
if we started on the extreme left again with a new value of 6 obtained by diminish- 
ing the previous value by y. Whenever the cycle is completed it is only necessary 
to subtract y from the old value of 6 to get the new one. The first value of 5 having 
been found as a—ie, it is simplest to write down the series of subsequent values of 6 
before commencing to draw the lines. Sooner or later, however, a value of 6 will be 
obtained which is less than yy, but as the addition of o to any argument leaves 5 un- 
altered it is immaterial whether we swbtract y or add (o —+), provided that we leave 
5<o. Hach new cycle after the first starts on the extreme left: the first cycle, as 
has been explained, starts on the vertical section corresponding to ic. 
A further modification is that of using (1+cos @) instead of cos @, so as to 
avoid negative quantities; the effect of this is that the constant 32D must be sub- 
tracted from the sum of the harmonic constituents. There are many advantages in 
the avoidance of negative quantities. 
For negative speeds (o negative) the changes necessary are as follows :— 
(1) Start with 7360°, 7360°+o, .... as headings to the vertical sections ; 
this gives a decreasing series of angles ; 
(2) the first value of 5 will be (7260°+io0—a), and readings will commence on 
the vertical section corresponding to 7360° + io ; the scale for 6 is always 
regarded as positive ; 
(3) the value of y, being positive, will be (3607 + vc). 
The only change required, apart from the construction of the abac, is that of the 
determination of 5; after the first cycle the procedure is the same as for positive 
speeds. 
A small-scale illustration of the abacs used is given in fig. 6 for the case 
o =37.4465°. At the top of the diagram isa horizontal scale for @ and cos @ to illus- 
trate the construction of the abac. As an example we shall take a=20°. Then the 
dotted lines in the upper figure correspond to 20°, 57.45°,...and cos @ can | 
then be read from the lower scale. The abac is drawn with overlapping sections f 
commencing at 0°, 37.45° ..., and covers 2x 360°. There are nineteen complete _ 
sections and one incomplete section. Hence v=19, 7=2, o=37.4465, whence 4 
7 = 720° —19 x 37.4465° = 8.5167°. 
The 6 scale is given for multiples of 10° in this illustration, and 1+ cos @ is given 
at intervals of 0.1, decimals being omitted; there are no negative values. With 
5 = 20° we get the following values of 1+cos @:— 
193, 154, 90, 33, 4, 12, 58, 121, 176, 200, 182, 180, 67, 17, 2, 26, 81, 145, 190, 
A new cycle is then necessary; this is given by 
5 = 20°—8.5167° = 11.4833°, 
and the values of (1+ cos @) are continued as 
HOG eUGT, LOD inte 
If, originally, we had a =132.34°, then we should look along the top of the abao 
for the nearest value of io which is less than a, in this case 112,34°, giving 5=20°; 
the readings would commence with 33, 
“* 
a 
