150 



The opinion generally adopted by chemists, that acids and alkalies 

 do not act upon resinous bodies, appears from this investigation to be 

 altogether erroneous; since the chief ingredient of lac which we have 

 seen is soluble in those menstrua, is now determined to be of a re- 

 sinous nature. 



Some hints are lastly given concerning the further uses that may 

 be made of these preparations in various manufactures, especially in 

 dyeing, and the preparation of colours : nor is it thought unlikely that 

 medicine may derive some advantages from the application of the 

 extensive series of acid and alkaline solutions of resinous substances, 

 which till now were thought to be unattainable. 



On the Integration of certain differential Expressions, with which 

 Problems in physical Astronomy are connected, fyc. By Robert 

 Woodhouse, A.M. F.R.S. Fellow of Cams College. Read April 12, 

 1804. [Phil. Trans. 1804,^. 219.] 



In the preamble to this paper the author states, that if the intro- 

 duction of the new calculi, as they have been called, has extended 

 the bound of science, it has also greatly increased its difficulties by 

 their number and magnitude : and that whilst the differential forms, 

 which can be completely integrated, occur only in few problems, the 

 investigations in physical astronomy give rise to differential expres- 

 sions which call forth all the resources of the analytic art, even for 

 their approximate integration. 



The main object of this paper is to give a method of computing 

 the integrals of certain expressions which lead to the determination 

 of the logarithms of numbers, and the lengths of circular arcs. In 

 treating of one of these expressions, known by the name of Fagnani's 

 Theorem, the author traces out the correspondence between the me- 

 thods of computation, and the proportion of geometrical figures ; the 

 analytical method, by which the integral expressing the arc of a 

 circle is computed, affording, when duly translated, the theorem for 

 the tangent of the sum of the two arcs expressed in terms of the 

 tangents of the arcs. 



It is in vain to attempt, without the use of symbols, to convey any 

 adequate, nay, even a faint idea of the various series, converging and 

 diverging according to the value of one of the coefficients of the 

 original expression, which lead to the conclusions that illustrate this 

 mode of investigation. Suffice it to say, that among other uses, the 

 method may be applied to expand the formula that occurs in esti- 

 mating the perturbation of planets : and in this instance the author 

 points out the series which would be most commodious, and which 

 would converge most rapidly if the radii of the orbits of the two 

 planets, whose perturbations are sought, were nearly equal. 



