ON MOLLUSCA OF THE NOETH-EAST ATLANTIC, ETC. 101 



parabola, the theory of elliptic integrals becomes simply common trigono- 

 metry, or parabolic trigonometry with the theory of logarithms. 



These views will suggest to us the reflection, how very small is the field 

 of that vast region, the Integral Calculus, which has hitherto been cultivated 

 or even explored ! When we find that the highest and most abstruse of 

 known functions, not only circular functions and logarithms, but also elliptic 

 integrals of the three orders, are exhausted, " used up," in representing the 

 symmetrical intersections of surfaces of the second order, who shall exhibit 

 and tabulate the integrals of those functions which represent the unsymme- 

 trical sections of surfaces of the second order, or generally those curves of 

 double curvature in which surfaces of the third and higher orders intersect ? 

 Considerations such as these but add fresh evidence to the truth, how small 

 even in mathematics is the proportion which the known bears to the 

 unknown ! 



Cheltenham, August 8, 1856. 



In revising this memoir for publication among the Reports of the British 

 Association, I have supplied several numerical examples to illustrate the theory. 

 I have added some new theorems, such as the curious properties of the 

 polygon of n sides circumscribing the parabola, p. 95 ; the theorem which 

 connects the corresponding points of the parabola and the equilateral hyper- 

 bola, p. 94 ; a new trigonometrical form for the roots of a cubic equation, 

 p. 81 ; and the geometrical expressions for the 2n roots of a trinomial equa- 

 tion, in the excepted case, by the help of parabolic trigonometry, p. 99. 

 I have also made a few other additions, and several corrections. — J. B. 



The Vicarage, Wandsworth, Nov, 10, 1856. 



Report on the Marine Testaceous Mollusca of the North-east Atlantic 

 and neighbouring Seas, and the physical conditions affecting their 

 development. By Robert MacAndrew, F.R.S. 



In the following Report, prepared in compliance with a wish expressed by 

 the Committee of the Natural History Section of the British Association at 

 the Glasgow Meeting last year, I have endeavoured to embody the results 

 of personal research, obtained principally by means of the dredge, at various 

 intervals during the past twelve years. 



The field of my labours has extended from the Canary Islands to the North 

 Cape (about 43 degrees of latitude), and with reference to the following 

 Tables, it should be explained that when a species is stated to extend north- 

 wards to the latter, or southwards to the former of these limits, it is not to be 

 inferred that it does not range further ; and this it is more important to bear 

 in mind, because a large proportion of the Mollusca inhabiting the coasts of 

 Finmark are known to be widely distributed in the Arctic Seas, while a con- 

 siderable number of the Canary species extend to, and in some cases attain 

 their maximum of development in, the tropical region. 



It is hardly necessary to add, that even within the district to which my 

 observations have been confined, many species of mollusca are recorded to 

 have been obtained which it has not been my good fortune to meet with or 

 identify, and that of all such I have taken no note. 



