498 TEANSACTIONS OF SECTION A. 



aud J„-i-4- a/ — = -l^i-i sin X-Q„_i cos X. 

 Hence, inserting these values in (16), 



.r 

 and l\i-i — P»+i = " Qh. 



Hence these expressions are independent of n. This result wa? due to 

 Mr. Gregory. 



To see whether they are also independent of a; I have evaluated 



PiPs + QiQ^ 



and find this expression = 1 ; also the first three terms of 1\X\ + Q,,Q|, leading to 

 the same result. 



We may conclude generally (by induction) that 



r«P„ , 1 + Q,.Q,H 1 = 1 . .... (17) 



Now, since tan ('„ = Q„-^P„ .ind P„ sec o„ = l^,„ (17) may bo written in either 

 of the forms 



1 + tan a„ tan a,, s i = ,, r> — ■ 



or 



sec (n„+i-n„) = E„'R„ii .... (18) 



These results may not be very important, but they are very simple and 

 interesting. The reasoning is inductive, and would need to be verified ; but this 

 would be easy if short tables of log P and log B were calculated as suggested 

 above, 



Department of Astronomy axd Cosmical PiiYsica. 

 The following Papers and Eeports were read : — 



1. Preliminary Note on the Rainfall rcriodoijrani. 

 By Professors A. Schuster, F.R.S., and H. H. Turner, F.R.S. 



The rainfall periodograms for Padua (170 years), Greenwich, and Klagenfurt 

 (90 years) show no marked periodicities from twenty-one months to five months. 

 There are, however, features deserving attention near five months in all three 

 periodograms, though the periods are not the same. For Greenwich the period is 

 five months, less one-fifteenth. It is not yet determined whether this is the actual 

 period of the wave or whether the use of monthly means converts a shorter 

 periodicity into this apparent one. A periodicity of 25-00 days is found to exist in 

 the Greenwich rainfall, but the coefficient is too small to give the one of longer 

 period by interference with the month. Other short periods are also indicated. 

 The results will be communicated in detail to the Royal Society. 



