TRANSACTIONS OF SECTION A. 443 
may be treated similarly, and so may other functions formed from the sine-function 
as f(a, p,) is from the exponential function. Among various generalisations 
which may be made I may mention the following :— 
If ©(u) is real and continuous throughout (0,1), the zeroes of the integral 
function 
1 
1 
1 
: | Ou” @(u)du (p>), 
Py 
which lie above the real axis, tend to the points 
fi) log (Qk) + log (=) + log at 
+i[ Qk +1)0 +37(1 +)|. 
6. Binary Canon Extension. By Lieut.-Col, ALLAN CunnincuaM, FF, 
The author has prepared a table showing the least Residues (R), both + and — 
of the powers of 2 (say of 2°), for all prime moduli (p), and also for all power of 
prime moduli (p*), up to p or p* > 10,000; the range of « is from « ¢{ the 
modulus up to 2 = 70. 
Thus it gives at sight the factors > 10,000 of (2*+ R), and of (2"+2*-* + 1) 
up to x = 70 (when Ris small), With some subsidiary work many other forms 
may be factorised, various congruences may be solved, and the Haupt-exponents 
(€) of small bases (a@) may be found, ze. the least values (€) giving af = + 1 
(mod p or p*). 
This table is an extension of the author’s Binary Canon (published in 1900) 
which extended only to moduli > 1000. 
7. On the Theory of Transfinite Numbers. By Dr. BE. W. Hosson, FAS. 
FRIDAY, AUGUST 19. 
SUB-SECTION oF ASTRONOMY AND CosmicaL Prysics. 
CHATRMAN—SriR Joun Extor, K.C.LE., M.A., F.R.S. 
The Chairman delivered the following Address :— 
When the suggestion was made to me that I should preside over this important 
sub-Section my first thoughts prompted me to decline the honour. The position 
had been filled during the past two years by two distinguished physicists, both of 
whom had dealt chiefly with the problems and the position of meteorological 
science, and hence I thought that it should be offered to some representative of 
cosmical physics. I also doubted whether an official meteorologist whose time has 
been chiefly given up to duties of administration could have anything of interest to 
communicate to you. However, on fuiler consideration it occurred to me that I 
might be able to place before you some features of Indian meteorology leading up 
to and assigning, as I hope, adequate reasons for the study of a portion of the field 
of tropical meteorology as a whole. 
My Address consists of three parts, viz. :— 
1. A brief sketch of the broad features of the meteorology of India in their 
relations to the general meteorology of the Indo-oceanic region. 
