THE STOCK OF WHALES 283 



viewed. Quite subtle atmospheric changes may make a great difference to the number 

 of spouts observed. 



In the second place the recoveries of whale marks might provide an independent line 

 of inquiry. If m equals the number of marked whales at large in a given area at the be- 

 ginning of the whaling season, m k the number of marked whales killed in the same area 

 during the same season, t the total whales killed in the same area during the same 

 season, and S the total number (to be ascertained) of whales at large in the same area, 



etc., then it might be argued that 



„ m x t 



m k 



on the ground that the ratio of marked whales killed to marked whales at large is as the 

 ratio of total whales killed to total whales at large. The difficulty is to assign proper values 

 to these symbols. The marked whales at large (m) might be calculated from markings in 

 previous seasons less previous recoveries, on the assumption that the whales return to 

 the same area after their winter migration ; but we do not know what wastage of marks 

 may have taken place, for a certain number must, for example, drop out of the whale. 

 The marked whales killed (m k ) must be greater than the actual number of marks re- 

 covered, for some marks must be overlooked or not sent in. It should be possible to 

 ascertain the number of whales killed (t), but the shifting nature of the population and 

 uncertainty of the limits of the area under consideration make it doubtful whether m 

 and t would be fair samples of S and properly comparable with each other. It might be 

 possible to apply such a calculation to one of the self-contained groups of Humpbacks, 

 but the unknown variables would still make such an estimate unreliable in itself. 



Hjort, Jahn and Ottestad (1933) have discussed the possibility of estimating the 

 magnitude of the stock of whales by a statistical method developed by Helland(i9i3-i4). 

 This method assumes a relation between the stock and the animals killed such that the 

 reduction of the number of animals killed to, say, one-half of the original total will mean 

 that the stock has dwindled to one-half. The basis of the calculation is that the rate of 

 reduction of the catches must be proportional to the fraction of the total stock killed each 

 year, i.e. if a large fraction of the whole stock is killed each year there will be a rapid 

 decline in the yearly catches and vice versa. Thus if A is the stock and B the catch in one 

 year, and p the percentage reduction of the catches in subsequent years, then A could 

 be ascertained from the formula 



A:B= 100: p. 



This method was applied by Hjort, Jahn and Ottestad to the diminishing catches in 

 the Iceland whaling from the peak in 1902 to its cessation in 191 5. As a check on the 

 results they also tried another mathematical method. They found that the catch per 

 catcher at first increased, reaching its maximum in 1895, and thereafter declined. The 

 estimation of the total stock was based on the assumption that when the yield per 

 catcher is at its maximum there is an equilibrium between the whales caught and the 

 rate of regeneration of the stock. From what is known of the reproductive habits of 



