126 PROFESSOR STOKES, ON THE DISCONTINUITY 



When the amplitude of x becomes 150° in place of 90°, the amplitude of ar^ is increased 

 by 180", Hence in the first series it will be sufficient to change the sign of the imaginary 

 part. To see what the second series becomes, imagine for a moment the factor uc put outside 

 as a coefficient. In the reduced series it would be sufficient to change the sign of the imagi- 

 nary part ; and to correct for the change in the factor oo it would be sufficient to multiply by 

 cos eo" +■%/- 1 sin 60°. But since the amplitude of w was at first 90", the real and imagi- 

 nary parts of the series calculated correspond respectively to the imaginary and real parts of 

 the reduced series. Hence it will be sufficient to change the sign of the real part in the 



product of the sum of the second series by B, and multiply by - (l + \/3 \/- 1), which 



gives the result 



- 25-30132 + 8-64681 \/- 1. 



Hence we have for the result obtained from the ascending series : 



for amp. a? = 90*, for amp. a? = 150", 



From first series - 20-14750 + 17-57548 \/- 1 - 20-14750 - 17-57548 \/- 1 



From second series + 5-16230 -|- 26-23499 \/- 1 -25-30132+ 8-64681 -s/- 1 



Total - 14-98520 + 43-81047 v/- 1 - 45-44882 - 8-92867 \/- 1 



On account of the particular values of amp. x chosen for calculation, the terms in 



the ascending series were either wholly real or wholly imaginary. In the case of the 



descending series this is only true of every second term, and therefore the values of the 



moduli are subjoined in order to exhibit their progress. The following is the calculation 



for amp. x = 90", in which case there is no inferior term. 



Coefficient 

 Real part. ofV— 1. 



+ 1-0000000 



+ 0-0086806 +0-0086806 



+ 0-0011604 



- 0-0001484 + 0-0001484 



- 0-0000563 



- 0-0000142 - 0-0000142 

 - 0-0000089 



+ 0-0000033 - 0-0000033 

 + 0-0000029 



+ 0-0000015 +0-0000015 

 + 0-0000017 



- 0-0000010 + 0-0000010 



- 0-0000007 - 0000017 



+ 1-0084677 + 0-0099655 v/- 1. 

 The modulus of the term of the order 12 is 14 in the seventh place, and is the least of 

 the moduli. Those of the succeeding terms are got by multiplying the above by the factors 



