BIOLOGY OF THE ATLANTIC MACKEREL 199 



by chance, produce a series of catch magnitudes conforming so well to hypothesis, 

 it has been proved not only that the hauls were quantitative, but also that the samp- 

 ling provided adequate representation of all parts of the distribution of each of the 

 various stages of eggs and larvae up to 22 mm. long. Nothing is yet proved as to 

 the extent of random variability, either of the quantities caught per haul or per cruise. 

 This would control the scatter of points in figure 17 and will receive consideration 

 in the final paragraphs of this section. 



The foregoing has dealt with the collection of material. Turning now to the 

 mathematical treatment, the initial step was to total the catches of a given stage for 

 each cruise and then average these totals for certain groups of cruises. This use of 

 total per cruise is equivalent to a direct arithmetic integration of the frequency surface 

 and could introduce no errors if the same stations were occupied on each cruise, and if 

 all stations represented equal unit areas. These requirements were approximately 

 met because the same station plan (fig. 14) was used for each cruise, and the stations 

 were distributed uniformly enough to represent approximately equal unit areas. 

 The principal change from cruise to cruise was the omission of some stations. As 

 earlier mentioned, stations north and east of Nantucket Shoals were omitted from the 

 first seven cruises, and it already has been pointed out that this probably had no 

 effect on the computation because these northeasterly stations could have contributed 

 nothing to the totals of the group of mackerel that is followed in the survival curve. 

 Besides this the stations at Martha's Vineyard IV, Montauk IV, New York V and 

 VI, and Cape May I were usually omitted. Since they proved always to be at the 

 periphery of the egg and larval concentrations, their exclusion or inclusion could make 

 little difference. However, on four of the first seven cruises, there were additional 

 omissions which could possibly have had important effects. 



On cruise I the station at New York I and all of those on the Montauk and 

 Shinnecock sections were omitted. Judging from the catches at adjacent stations, 

 and also from the distribution of appropriate stages on the following cruise, tliree of 

 these omitted stations might have added low to medium catches to the totals for 

 stage A and B eggs, but this could not have increased their totals for that cruise by 

 more than 5 percent, and could have modified the average per cruise of the four 

 cruises used for these stages by less than 2 percent, so the effect of this omission is 

 inappreciable. 



On cruise IV all stations on the Winterquarter section, and those at Chesapeake I 

 and III were omitted. This omission would have a serious effect on the total for that 

 cruise, for these stations could have been expected to yield nearly maximal numbers of 

 4- to 8-mm. larvae, but the effect of this omission was rectified by substituting the 

 cruise III values for these stations in calculating the average per cruise. (See foot- 

 note p. 192.) This substitution could have introduced error only to the extent of 2 days' 

 growth and mortality — an effect that would not be perceptible after inclusion of the 

 data for the three other cruises in the group average. 



On cruise V the stations on the Martha's Vineyard section, at Montauk III, and 

 at Shinnecock I and II were omitted. This probably reduced the totals of 3-mm. 

 larvae appreciably, and 4-mm. larvae slightly. If the effect on the 3-mm. larvae is 

 estimated by examining the result of substituting numbers interpolated from the 

 previous and subsequent sampling at these stations, the total for this size of larvae is 

 increased from 5,215 to 12,549 for cruise V and the average per cruise for cruises II 

 to V is increased from 9,310 to 11,144. Substituting the latter in table 7 and carrying 

 the work through to the logarithm of the empirical number surviving per million, it is 



