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Determination of the North Polar Distances and proper Motion of 

 Thirty fixed Stars. By John Pond, Esq. Astronomer Royal, F.R.S. 

 Read June 15, 1815. [Phil. Trans. 1815, ;>. 384.] 



In the former catalogue which the Astronomer Royal gave in 1813, 

 he estimated the probable extent of error at not more than one fourth 

 of a second ; and his present catalogue, which may be presumed to 

 be improved by a greater number of observations, confirms the just- 

 ness of that estimate ; since the greatest difference observable is not 

 more than two tenths of a second. 



A comparative catalogue is also given of the places of the same 

 stars in 1756, as deduced from the observations of Dr. Bradley ; and 

 thence is added a column of annual proper motions for each of the 

 stars in the collection. 



An Essay towards the Calculus of Functions. By C. Babbage, Esq. 

 Communicated by W. H. Wollaston, M.D. Sec. R.S. Read June 15, 

 1815. [Phil. Trans. 1815, p. 389.] 



In the same manner as an exponent expresses one operation on 

 quantity, namely, the multiplication of it by itself a certain number 

 of times, or raising it to the power expressed by that exponent, so 

 the term Function, which has been introduced into modern analysis, 

 is intended to express generally the results of all the various opera- 

 tions that can be performed upon quantity. Many of these operations 

 consist of two parts, the direct and the inverse. To extract a root is 

 the inverse, with reference to that of raising any number to a higher 

 power. So the integral is the inverse of the differential calculus ; and 

 the same observation applies to finite differences. In all these cases 

 the inverse method is by far the more difficult of the two. 



The author's object in the present essay is to consider the inverse 

 method with respect to functions, and, if possible, to determine the 

 value of an unknown function by means of any functional equation 

 given, instead of discovering, as in the direct method, the value of a 

 quantity from an equation in which the function is known. But the 

 mode in which the author pursues his inquiry, of course could not 

 admit of being publicly read. 



Some additional Experiments and Observations on the Relation which 

 subsists between the Nervous and Sanguiferous Systems. By A. P. 

 Wilson Philip, Physician in Worcester. Communicated by Thomas 

 Andrew Knight, Esq. F.R.S. Read June 15, 1815. [Phil. Trans. 

 1815, p. 424.] 



This paper comprises a series of very numerous experiments on the 

 effects of various stimuli applied to the brain and nerves of rabbits 

 and frogs, in exciting the voluntary muscles, the heart, and the blood- 

 vessels ; from which the author infers, 



