

WITH AN APPENDIX BY PROFESSOR A LODGE. 



105 



TABLE showing the Acute Angles (in degrees) whose Cosines were required in 

 Forming the Terms belonging to the Harmonics of Even Order. The Signs pre- 

 fixed to the Angles Indicate whether their Cosines had to be Added or Subtracted. 



Note. The terms in each of the columns headed 40, 80, 120, ... all balance, their 

 sum being zero. 



The terms in each of the columns headed 20, G0, 100, ... balance except the 

 last term. 



Hence the sum of all the 18 values of P (0), from 5 to 90, 



This equivalence was made use of in finally checking the values of the harmonics of 

 each even order. 



Some Cliaracteristies of the Functions as shown by the Tables. 



The functions become more and more undulating as n increases, P(0) having n 

 zeroes between 0=0 and 180, similarly spaced on either side of 90. The most 

 remarkable peculiarity noticeable in drawing their graphs is that the intervals 

 between the successive zeroes from the first to the "' are almost exactly equal. 



The graph of P 20 is reproduced in fig. 2, to emphasize this peculiarity of equal 

 intervals. 



VOL. com. A. r 



