of the Fishery board for Scotland. 



211 



jaw to the end of the opercular bone, on the upper side." ( Kving to the 

 indefiniteness of the latter point, and to the consequeut difficulty of taking 

 this measurement accurately, this dimension is not so good or reliable as 

 that taken on the hard skull. This dimension was measured along 

 the ventral aspect from the extreme anterior edge of vomer to the extreme 

 posterior edge of the basioceipital. As with the body-height, it is 

 expressed in percentage of the total length minus the tail and length of 

 skull — i.e., of the " body-length." 



Secondary sexual differences are somewhat more evident in this dimen- 

 sion than in the case of the body-height. In all five groups, Table XIII., 

 p. 232, shows that the averages for the males are smaller by *73 on an average 

 than those for the females, although in only two groups — viz., the 

 mature specimens from the southerly North Sea — are the fluctuations of 

 the averages free from overlapping. If the same process of calculating 

 the probability of the results be gone through here as with the body- 

 height, it is found that the fluctuations of two more groups do not overlap 

 when three times the probable error is employed. Heuce, for these two 

 groups the probability of differences existing is 19 to 1. For the fifth 

 group — the mature specimens from Aberdeen — the fluctuations of the 

 average cease to overlap when twice the probable error is employed, and 

 the probability is then 4 to 1. In conclusion, therefore, the probability 

 is, on the whole, in favour of there being secondary sexual differences 

 in this character — being 1000 to 1 for two groups, 19 to 1 for two other 

 groups, and 4 to 1 for the fifth. If an average be struck between these 

 probabilities it may be said that the probability is about 100 to 1 in 

 favour of there being secondary sexual differences. 



Two fine points in connection with the working of the method may be 

 mentioned here. It has been shown that the group of mature specimens 

 from Aberdeen were not very representative with regard to the body- 

 height, and this conclusion might be held to vitiate the value of this 

 group in all other characters. This, however, is not so, because the body- 

 height is an extremely variable character, having a wide range of 

 deviations, and is, further, not so true and easily measured a character as 

 the leugtli of skull. Hence, the conclusions formed with regard to 

 the body-height of this group do not apply to other characters. 



Again, this group was at variance with the other groups in regard to 

 the body-height, but in regard to the length of skull it is in entire 

 agreement with the others — the greater size of the probable errors and 

 thence of the fluctuations of the averages being really due to the smaller 

 number of specimens. This example shows well how the probability or 

 certainty of our conclusions varies with the number of specimens examined. 



It may be mentioned that the only dimension which Heincke 

 found to vary with the sex was the side length of the head, and the 

 difference, in favour of the females as above, was very small, only *5°/ 00 

 (Lc, p. 94). 



In the other dimensions of the skull, skull-breadth and skull-depth, 

 Tables XIV. and XV., pp. 233, 234, show that relative to the skull-length 

 there is no sex- variability. The differences in the averages for the males and 

 females are very slight, and sometimes the average for the male is the 

 higher, sometimes that for the female. 



It has just been seen, however, that the skull-length is slightly greater 

 in the female than in the male. Hence, the measurements of the skull- 

 breadth and skull-depth partake of this difference when referred to the 

 body-length. The calculation of the exact amount of difference is some- 

 what complicated, but if approximate values are given to the breadth and 

 depth of skull a very close approximation to the true values can readily 

 be obtained. From the Tables it will be seen that the skull-breudth is 



