104 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
We have seen that when such a linear relationship holds, the graph for 
the “ take-np ” is also rectilinear. The equation to the flowing curve, which 
lies most evenly amongst the points of fig. 5, is : 
n= *022(0- 12) 2 + 5-6 
= *022C 2 - *53C + 8*8 . . . . (9) 
This equation does not hold beyond the limits 12’s and 40’s, and it 
will be seen from fig. 5 that - even between these limits the graph does not 
represent closely the relationship between the plotted points. Substituting 
this value of n in the expression for the “ take-up,” 
y = (' 83# 2 - 2’93£)A+ 'I77x - where A = i . 
jA. 
Let us consider the cases when x — 10 and x— 12 ; 
then, y = 53'7A + 1*77 - -5^-? when ^=10; 
-A. 
y = 84*0A 4- 2T2 - when a; =12. 
A 
On examining the above expressions for y we see that they can be 
broken into two parts, y x and y 2 : 
y^ = 53*7A + T77 \ 
where *073 Vwhena;=10 
v *-~r ) 
and y = yi~y 2 ; 
y 1 is an equation to a straight line, 
and y 2 is an equation to a rectangular hyperbola. 
Also, we are tempted to say that when A is very large, the graph for the 
“ take-up ” approximates to a straight line, and similarly when A is small, 
the “ take-up ” graph approaches hyperbolic form. This assumption would 
be quite erroneous, because the equation is only known to represent the 
actual conditions between the limits, *025 and ‘0833 for A. Therefore A 
can neither be very large nor very small without violating the conditions on 
which the equation was deduced. 
From equation (9) we calculate that for 50’s yarn (i.e. for A = - 02), n, 
the twist should be 3 7 3 turns per inch — a proposal which no spinner could 
entertain. 
Fig. 7 shows the synthesis of the graph for x = 10 and its relation to 
the graph exhibiting the experimental results for ten turns of twist. The 
latter is lettered B in the diagram. 
It is always possible to obtain two straight lines which approximate 
closely to a curve between limits. From fig. 8 we see that for the case 
