TRANSACTIONS OF SECTION A. 637 
having a speed greater than that of the wind? (5) Different periods of residual 
swell being, apparently, characteristic of different localities, will selective absorp- 
tion cause waves of this period to be developed at a specially rapid rate when a 
wind blows in this or neighbouring areas? (4) What is the relation between 
growth of wave-length and diminution in curvature of wave-front ? 
Further, an examination is made of the numerical relations among the quan- 
tities recorded in Lieut. Paris’ paper on deep-sea waves,! and in Antoine’s collec- 
tion of results,? and in Coupvent Desbois’ summary of the observations of 
amplitude made on the Astrolabe. I find relations among Paris’ numbers which are 
useful for the interpretation of his observations. I doubt if Antoine is justified in 
applying Desbois’ formula H o V# to later observations than those of the Astro- 
labe. Paris’ observations seem, however, to be pretty nearly comparable with 
Antoine’s collected observations, of which, indeed, Paris’ form part. Antoine, 
assuming from Desbois that H o V3, finds L o V#, and therefore that the 
‘modulus’ HL oc V"", for what he records as fully developed waves. He thinks 
that this ‘modulus’ HL may ultimately be found proportional to V, with H 
remaining, I suppose, proportional to V? and L becoming proportional to V3. 
He finds Desbois’ empiric relation H cc Vi ‘ readily justified’ on theoretic grounds. 
I think this supposed theoretic justification is illusory, and that it has hindered 
Antoine from obtaining the best numerical relations from the data at his disposal. 
I consider that his numbers are better represented by taking H proportional to 
V? and L proportional to V+, which gives the relation between L and H suspected 
by Antoine. These formule are statistical, not dynamic. Their value depends on 
the number of observations. H can only be regarded as varying with V! when we 
average H and V over a large area. Desbois’ table of amplitudes in ‘Comptes 
Rendus,’ xii. pp. 82-87, seems to indicate that a large fraction of the total turbu- 
lence of many parts of the ocean is due to their invasion by swells from a distance. 
From an examination of Paris’ numbers I find that—(1) The average steepness 
of the waves increases with excess of V (velocity of wind) over U (velocity of 
wave) ; (2) The law that persistence of amplitude (in time) is proportional to L? 
is recognisable ; (8) That, even far from coasts, geographical position modifies the 
law in this way, that, for the same average wind velocity, the average amplitude is 
greatest in the Southern Ocean, z.e. south of the Cape of Good Hope and Cape 
Horn, where the wind blows always from the north-westward and the wave- 
pulses are free to chase one another round and round the globe. Perhaps there is 
also some increase, due to focussing. Further, the specific roughness in the 
regions of Indian Trades, Atlantic Trades, and Western Pacific is in the order of 
their exposure to the Southern Ocean, which is the order in which they have been 
here named. They may be regarded as branches of the Southern Ocean. The 
Western Pacific is greatly sheltered from the Southern Ocean by a screen of 
islands. Paris’ observations do not extend to the Eastern Pacific, as do those of 
Desbois. The latter show that the specific roughness is much greater in the 
Eastern than in the Western Pacific. In the semi-closed Seas of China and Japan 
the specific roughness is less than in the above oceans. 
SATURDAY, SEPTEMBER 16. 
1. On the Existence of Masses Smaller than the Atoms.4 
By Professor J. J. Tuomson, 2.2.8. 
! Revue maritime, ¥xxi, 
2 Sur les Lames de Haute Mer. 
8 See p. 8 of Les Lames de Haute Mer. 
* Published in the Phil, Mag. Dec. 1899, pp. 547-67. 
