[cokek] determination OF THE FORM OF CAINIS S7 



A series in wiiich the convergence depends upon the values of the sine 

 terms, since the sum of the coefficients is obviously divergent. On examin- 

 ation the maximum value of the series is found to be 



n = 1 n 4 



and the series is semi-convergent and may be integrated. 



The integration produces a series of the same form as the previous 

 example and hence the displacement cam can be drawn. 



Owing to the simplicity of the instances the resulting displacement 

 curve could be determined more easily by other methods, but if the 

 periodic curve is of more complicated form this would not be the case. 

 For instance, the curve to be analyzed may be made up of arcs of curves 

 of a higher order of single valued functions where it may be difficult to 

 evaluate the definite integrals involved. A graphical process may 

 then be resorted to, as indicated by Professor Clifford,^ 



This may be briefly described as follows : If the periodic curve to 

 be analysed be wrapped round a cjMinder bO that, without altering the 

 value of the ordinates, it completely encircles the cylinder m times, then 

 it can be shown that the orthogonal projection of this curve on the 

 meridian plane, which passes through the zero point of the curve, will 

 enclose an area which is proportional to the coefficient J„j, and, on a 

 plane at right angles the orthogonal projection will be proportional to i?„i. 

 Other methods have been described, such as that described by Langsdorf^, 

 which merely involves the construction of lines and circles. 



In general, therefore, it is possible to determine the displacement 

 curve when the accleration or velocity is prescribed. 



It may also be noted that sine and cosine functions are not the only 

 ones into which the original curve can be analysed, for a function of x 

 can be expressed in terms of a series of zonal or cylindrical harmonics 

 in which the co-efficients a>-e expressible in terms of definite integrals, as 

 in Fourier's series. 



For ordinary purposes there is no advantage in these forms, as the 

 series becomes extremely complicated, and graphical methods of obtain- 

 ing the coefficients are more or less undeveloped. 



1 Collected papers p. 201. 



2 Langsdorf— A graphical method of analysing distorted alternating current 

 waves. Physical Review, 1901, p. 184. 



