498 Proceedings of the Royal Society 
the problem becomes this, — “ To draw upon the surface of the 
vault a curve which shall cross all the lines of pressure squarely/’ 
This belongs to a well-known class of problems in what is called 
the calculus of variations. 
The nature of this curve must depend on the character of the 
arch ; yet it has certain general features independent of that char- 
acter. The chief of these may be explained in this way. If we 
take two closely contiguous curves, inclosing between them, as it 
were, a course of arch-stones, the breadth of that course, at any 
place, is proportional to the cosine of the inclination of the line of 
pressure. Hence, in every skewed arch the breadths of the stones 
as seen on the parapet plane, must diminish from the crown down- 
wards, becoming at 60° from the crown just half as broad as at 
the top. 
In the case of the circular arch, the projection of the curve upon 
the plane of the parapet is the well-known tractory, which is 
asymptotical to the horizontal line passing through the centre. 
Hence we cannot have a semicylindric skewed arch, because the 
curve of the course-joint cannot reach to the vertical part of the 
surface. 
The nature of the true arrangement is shown on the other side 
of the model. 
A glance at the ends of the arch-stones of any skewed bridge is 
enough to apprise us of whether or not the structure have been 
properly arranged. 
3. On the mode of Growth and Increase amongst the Corals 
of the Palaeozoic Period. By H. Alleyne Nicholson, M.D., 
D.Sc., Professor of Biology in the Durham University 
College of Physical Science. 
In the first portion of this communication, the author discussed 
the general phenomena exhibited by the Palaeozoic corals as 
regards their mode of growth and increase. Five chief modes of 
growth were distinguished : — 
