412 SECTIONAL TRANSACTIONS.— A. 
The study of the stratification of coastal waters has been carried on for 
nearly half a century by Swedish oceanographers, and important relationships 
between the displacements of the water-layers and the migration of the herrings 
and other food-fishes have been discovered in the course of this work. The 
dynamics of these movements, and notably the vertical displacements of the 
boundary surfaces, which occur on a vast scale along the west coast of Sweden, 
have been made the subject of continuous observations by means of instruments 
specially designed for this purpose, sbme of which instruments were shown 
by the lecturer. Making use of the correlation method, the author has been 
able to prove that a distinct influence is exerted by the wind on these internal 
movements, whereas the air-pressure appears to be void of any similar effect. 
The influence of cosmical factors, again, has been thoroughly investigated by 
Otto Pettersson. 
The importance of these internal movements on the hydrography of the 
Baltic is very great. The bottom water in its deeper basins being too dense 
to become ever able to rise to the surface through a surface-sheet of low-salinity 
water forty to sixty métres thick, its supply of oxygen would gradually become 
exhausted and the water made uninhabitable for its marine fauna, if there was 
not an intermittent inflow of freshly saturated sea-water in the shape of an 
undercurrent through the straits taking place at intervals largely ruled by the 
occurrence of the internal movements before mentioned (experiment). These 
invasions of the undercurrent make the impression as of submarine waves 
sending gigantic ‘ breakers’ over the thresholds to the Baltic and gliding down 
the slopes leading to the deeper basins like submarine streams or rapids. 
(c) Department oF MarHeMATIcs. 
23. Rev. J. Cunnen.—The Identity 4X =Y¥? —( — 1)-p Z?. 
Ifp be any odd prime and X=(x? —1)/( —1) it is known that 4X can be expressed 
as above. Legendre, in his ‘ Théorie des Nombres,’ stated erroneously that Y may be 
determined by expanding 2(zx —1)2? ) by the binomial theorem, and reducing each 
coefficient to its absolutely least residue (mod. p). He afterwards, however, corrected 
his mistake. 
From the correspondence in Nature, of June 9 and July 7, 1921, together with 
results given in Mathews’ ‘ Theory of Numbers,’ p. 218, it appears that all odd primes 
p<4l1 conform to Legendre’s rule, while those from 41 to 61 inclusive fail. It seems 
therefore that the extent of failure has not yet been completed, and the object of the 
present paper is to effect this. 
Let p=2p'+1 and Y=2x*+ea?!- 14... tepa!-"4+ 2... «then 3.2%;—189 
+(90 + 36e2 + 32e3 + B2ey¢3)e:p + (1 + 4ea)p2, where e:x=(—1)?,  1e=(2/p),  & 
=(3/p) (Mathews, p. 217). The condition in this case for the failure of Legendre’s rule 
is /¢e;/>4(p—1l) and if p=«a-+t 24¢ all primes > 3 are included in this form. Where 
a=1, 5,7, 11, 13, 17, 19, 23, the e’s are determined by the a’s, and the result of sub- 
stituting for p is 8 simple quadratic expressions inf forc;. I find the condition for 
failure /c;/ > 4(a —1)+-12t is satisfied if t exceeds any of the values below :— 
Og Td LS LT, 19) 23: 
CPUS 1 ORO alee) 
The only prime unaccounted for is 71, and on working out Y in this case, I find 
the term 582. Hence, Legendre’s rule fails for all primes p> 37. 
24. Dr. F. E. Hacxerr.—Lhe Relativity Contraction in a Rotating 
Shaft moving with Uniform Speed along its Azis. 
This problem was considered in terms of a fixed ether and the Fitzgerald- 
Lorentz contraction (A). The restricted principle of relativity (B) was applied 
to observations made by a fixed and a moving observer. The validity of 
expressing the results of their observations in terms of Euclidean geometry was 
assumed throughout the paper (C). 
A hypothetical modification of Fizeau’s method for measuring the velocity — 
of light was considered—a rotating shaft carrying two discs with apertures 
which correspond to the toothed wheel in Fizeau’s experiment. It follows 
readily that when a rotating shaft is moving with uniform velocity along its 
