220 Proceedings of Royal Society of Edinburgh. [sess. 
reduced. That is to say, without any further proof it is clear that 
AGCHE cannot he that particular S.H. curve to which the actual 
trace most closely approximates. 
Plainly the proper procedure in this problem is first of all to 
find the particular S.H. curve to which the trace ABODE 
most nearly conforms, and then to ascertain the greatest difference 
between the two curves, which quantity will be the required 
maximum deviation of the trace. 
Fig. S. — To keep the two curves sufficiently apart in the above figure , the 
deviation of the curve ABODE has been very greatly magnified, by 
the device of drawing the latter curve as if the length of 1 in the 
machine were only 2*5, instead of 30. 
The distance between the fixed point R in fig. 6 and the pen 
which we have supposed to be attached to the free end of the 
wire led down from Q, if measured up to Q and along QE, is 
given by the expression 
m - J(l 2 + r 2 - 2 rl cos t) . . . (4) 
where l = RO, and r = OS as before, and where m is the total length 
of the wire from the pen to the point S. 
The distance from R, also measured up to Q and along QR, of 
an imaginery point which moves with any S.H.M. whatever in 
the same vertical line as the pen, is given by the expression 
w + p cos (qt + d) .... (5). 
