340 
Proceedings of the Royal Society 
so that the solution of the problem resolves itself into that of this 
trigonometrical question, “ To find two circular arcs in a given ratio 
and of which the tangents are in the same ratio T 
The solution of this question, again, resolves itself into that of 
algebraic equations of the order ^(a + ^8 - 3) or J(a + /3 - 4), when 
a and jS are represented by integer numbers prime to each other. 
A table of the solutions for all possible values of a and (3 up to ten 
is given ; and it is remarkable that these solutions apply whether 
the motions of the two arms be in the same or in opposite direc- 
tions. 
The problem is then extended to the case of the common cycloid 
in which the centre of the revolving wheel moves along a straight 
line. Here it is shown that we have to discover those arcs which 
are equal in length to their own tangents ; and that the revolving 
arm must be made proportional to the secants of these arcs. A 
table of the first ten solutions is appended to the paper. 
The following Gentlemen were duly elected Ordinary 
Fellows of the Society. 
Alfred R. Catton, B.A. 
Rev. Francis Redford, M.A. 
The following Donations to the Library were laid on the 
table. 
Descriptive Catalogue of the Pathological Specimens, contained in 
the Museum of the Eoyal College of Surgeons of England. 
London. Supplement II. 4to. — From the Council of the College. 
The Stereoscope, its History, Theory, and Construction, with its 
application to the Fine and Useful Arts, and to Education. By 
Sir David Brewster, K.H., D.C.L., &c. London, 1856. 8vo. 
— From the Author. 
The Kaleidoscope, its History, Theory, and Construction, with its 
application to the Fine and Useful Arts. By Sir David 
Brewster, K.H., D.C.L., &c. Second Edition. London, 1858. 
8vo. — From the Author. 
A Treatise on New Philosophical Instruments for various purposes 
in Arts and Sciences, with Experiments on Light and Colour. 
