222 
Proceedings of Royal Society of Edinburgh. [ses?.. 
less than ST, but PQ' will be greater than PQ. Therefore the 
maximum discrepancy that now exists between the curves APBSC 
and DQETF must be greater than before ; for if PQ' be still the 
maximum discrepancy, it is greater than PQ ; and if PQ' be not 
now the maximum discrepancy, then the latter — whatever it be — is 
greater than PQ,' and hence a fortiori greater than PQ. Therefore 
the maximum discrepancy between the actual motion and an 
isoperiodic S.H.M. of any amplitude is always least when the two- 
motions are cophasal. As this result holds for any amplitude of 
the comparison it must hold for that particular one which 
approaches most nearly to the actual motion. Therefore we may 
put a = 0 in equation (5) ; which thus finally becomes 
iv + p cos t ; 
and writing 
w — m=v . . . . . (7) 
(6) then becomes 
e =p cos t + v+ J (l 2 + r 2 - 2 rl cos t). 
In this equation e, the error of motion, is a function of three- 
independent variables, namely t, jy, and v. 
First of all to deal with the occurrence of t in the above equation. 
Reference to fig. 6 will show that whatever succession of valuer 
e assumes as S in the course of its rotation passes down from its 
highest to its lowest point, e will again assume in reverse order as 
S returns up once more to the highest point. That is to say, we 
need only investigate the values of e, where t ranges from 0 to tt 
(inclusive), all other values of t giving mere repetition in tho 
values of e. 
Then again we are only concerned to find the largest error of 
motion committed, and this must occur at the time when 
0e/0Z = O .... (8) 
But 
de . , rl Sin t 
dt 1 v/(/ 2 + r 2 - 2rl cos t) 
hence from (8) either 
sin t — 0 or 7r, ^ 
