548 Royal Society : — 



■ In the same letter Mr. Babbage refers to the following docu- 

 ments : — 



Extract from a letter of Mr. Babbage to Sir H. Davy, 3 July, 

 1822, printed by order of the House of Commons. No. 370, 

 1823 : — 



" Another machine, whose plans are more advanced than several 

 of those just named, is one for constructing tables which have no 

 order of differences constant (p. 2). 



" I should be unwilling to terminate this letter without noticing 

 another class of tables of the greatest importance, almost the whole 

 of which are capable of being calculated by the method of dif- 

 ferences. I refer to all astronomical tables for calculating the places 

 of the sun and planets. It is scarcely necessary to observe that the 

 constituent parts of these are of the form a sin 6." (p. 5.) 



He refers also to an extract from the Address of H. T. Colbroke, 

 Esq., President of the Astronomical Society, on presenting to him 

 the first medal given by the Society, 1824; and to a description of 

 his machine by the late JVIr. Baily, published in Schumacher's 

 ' Astronomische Nachrichten,' No. 46, and republished in the ' Phi- 

 losophical Magazine ' for May 1824, p. 355. This last paper de- 

 scribes fully what could be done by the new contrivance. 



I have ventured to insert this postscript without consulting my 

 colleagues, as it is desirable not to delay the publication. 



G. G. Stokes. 

 London, Oct. 5, 1855. 



December 6. — Sir Benjamin Brodie, Bart., V.P., in the Chair. 

 The following communication was read : — 



'• On the Representation of Polyhedra." By the Rev. Thomas P. 

 Kirkman, A.M. 



This paper constituted an addition to the paper by the same 

 author read June 21, 1855. 



The author observes that to every p-acral q-edron corresponds a 

 p-dral q-acron, the summits and faces of either having the same order 

 and rank as to the number of edges with the faces and summits of the 

 other. When p = q, the corresponding pair will sometimes be iden- 

 tical figures, as to the number, rank, and arrangement of faces and 

 summits; and at other times, as is always the case if^ be not equal 

 to q, the two figures will differ. When they differ they may be 

 called a sympo^ar pair, both being heteropolar ; when they form one 

 and the same figure it may be styled an autopolar polyhedron. An 

 elegant way of representing a sympolar pair is deduced from the 

 two following theorems : — 



A. The q summits of a q-acron are the angles of a closed polygon 

 of q sides, all edges of the q-acron. 



B. A closed polygon oi p sides can be traced on the p faces of 

 every p-edron, having a side in every face, and passing through no 

 summit. 



