464 Mr. J. B. Dale on the 



this number can be found only by trial. Supposing, for 

 example, that three periodic terms are involved, then it must 

 be possible to find constants p l9 p 2 , pz such that the relation 



Wa r +p 1 Wa r +p 2 Ea r +p 3 a r = . . . (4) 



is identically satisfied for all values of r. 



In any actual problem, on account of errors in the data it 

 is impossible to satisfy this relation accurately ; but if the 

 residuals obtained on substituting successive values of a r in 

 the expression are always small and exhibit no s} T stematic 

 run, we may regard the assumption of three independent 

 periodic terms as justified. 



In the problem considered in the present paper it is found 

 that the assumption of two periods satisfies this condition, 

 and the relation to be satisfied is 



Wa +piEa r +p 2 a r =0. . . . . (4') 



The values of p v p 2 are found by solving two of the 

 equations. 



With the values of the p's thus found, the equations 



z z -j-p 1 z 2 -\-p 2 z+p 3 = (5) 



or z 2 +p x z + p 2 = 0, . . . . (5') 



as the case may be, are formed and solved. 



Considering (5), if the roots are z x , z 2 , z s , then the three 

 speeds T , 6 2 , #3 °£ t° e periodic terms are given by 



^ 1 = cos" 1 ^1, # 2 — c°s -1 iz 2 , 3 = cos -1 \z z . . (6) 



Finally, to obtain the amplitudes and arguments corre- 

 sponding to <9 i? we form the series of quantities 



P r 1 = E 2 a r — (z 2 + z. s )Ea r + z 2 z 3 a r . ... (7) 



Then c 1 sin(6' 1 r + ai )=( p V-i- :p V)Si, ... (8) 



Cl cos(^ 1 r + a0 = (PV-i-fPV)O 1 , . . (8') 



where , (\ = 1/2 sin X (z ± - z 2 ) (z x — z 3 ) , 



8^1/(2-^)^1-^)^1-^) .-. (9) 

 In like manner the arguments and amplitudes of the other 



periodic terms are obtained. 



3. The curve here submitted to analysis is part of that 



taken at Pulkowa, Sept, 18, 1910, and reproduced as 



Plate 7 A in Walker's book. 



By means of tracing-paper ruled in millimetre squares, 



readings of the ordinates were taken at intervals of 1 mm. 



