Correspondence on various mathematical theories and calculations, mainly pure, but some applied, and miscellaneous papers on scientific matters. The correspondents include, among others, A. de Morgan, C. Pritchard, P. Kelland, Mr Plarr, W.R. Hamilton, G.P. Bond, W.H. Miller, Lord Wrottesley, W. Whewell, J. Tyndall and J.C. Jamin. There are papers on the theory of probable errors; different forms of tabular interpolation; indirect demonstration; the quadrature of the circle; the political economy; and the sounds produced by the combustion of gases in tubes. There are also papers by Airy: 'On the substitution of methods founded on ordinary Geometry for methods based on the General Doctrine of Proportions, in the treatment of some Geometrical Problems'; 'On Partial Differential Equations of the 1st and 2nd Orders Homogeneous and with Constant Coefficients'; 'Suggestion of a Proof of the Theorem that every Algebraic Equation has a Root'; and an investigation of d[Theta]/de in the planetary theory.