118 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



Iii order to make comparisons easier the ratios ex- 

 pressing the distance of the dorsal hn from the tip of the 

 snout are divided into four groups each of which is 

 distinguished by a number: — 



1. When the ratios varied from 2'08 to 2*17 

 2 2-18 to 2-27 



3. „ „ „ „ 2*28 to 2 - 3T 



I. „ „ „ „ 2-38 to 2-47 



In a similar manner each of the other ratios was divided 



into four groups with the exception of the distance of the 



anus whicli was divided into five. To each of these groups 



descriptive numbers or letters were applied, viz. : 



Distance of Ventral Fin. 



Distance of Anus. 



Length of Anal Fin. 



a. 1-97 to 2-04 



b. 2-05 ,, 2-12 



c. 2*13 „ 2-20 



d. 2-21 „ 2-28 



1-41 to 1-45 



I. 1-46 „ D50 



II. 1-51 „ 1-55 



III. 1-56 „ 1-60 



IV. 1-61 ., 1-65 



A. 

 B. 

 C. 

 1). 



80 to 92 



93 „ 105 



10C) „ 118 



119 „ 131 



In the first place it should be noted that the formulae 

 do not express what they are intended to, and thai ii i< 

 quite possible that they can convey false impressions. 



For instance it is quite possible that two groups of 

 spring herring, to each of whicli ffeincke's average 

 formula 2bII. applies, are really much more unlike one 

 another than one of them is unlike a group of autumn 

 herring to which lleincke's formula 2al. applies. 



This difference may theoretically be very great, as may 

 be. seen on reference to Ihe following table: 



Formula . 



2bII. Spring herring 

 2al. Autumn herring 





Indices. 





D. 



2-18 

 2-27 

 2-27 



V. 



2-12 

 2-05 

 2-04 



A. 



1 '55 

 1-51 



1 -50 



