69 



diately perceptible. After stating the methods for adjusting this 

 thermometer for the measurement of the greatest heights, the author 

 details some experiments upon altitudes made with an instrument, 

 552 parts upon the scale of which were equal to 530 feet in altitude. 

 With this instrument boiled on the counter of a bookseller's shop in 

 Paternoster-row, estimated between four and five feet above the foot 

 pavement on the north side of St. Paul's Churchyard, and boiled 

 again in the gilt gallery of the cathedral, there was a difference of 

 254 parts ; the corrected height thus indicated therefore = 27 2 "64 

 feet. General Roy makes the gallery above the north pavement to 

 be 281 feet, which, allowing five feet for the difference of station, 

 brings the author's estimate to 267 feet, differing only four feet ; or 

 by another calculation, founded on General Roy's statement, the dif- 

 ference is less than two feet. 



Observations on the Analogy which subsists between the Calculus of 

 Functions and other branches of Analysis. By Charles Babbage, 

 Esq. M.A. F.R.S. Read April 17, 1817. [Phil. Trans. 1817, 

 p. 197.] 



At the commencement of this paper the author states the advan- 

 tages which may be derived from the employment of analogical rea- 

 soning in mathematics, and recommends it as a very useful guide to 

 new discoveries : he then proceeds to point out the striking resem- 

 blance which subsists between several parts of common algebra and 

 the integral calculus, and similar parts of the calculus of functions. 



Mr. Babbage then notices certain fractions which, by peculiar re- 

 lations among the functions of which they consist, become evanescent. 

 The true values of these fractions are ascertained, and they are ap- 

 plied to the solution of a class of functional equations which the 

 author had solved in a former paper, from which the following result 

 is obtained : " Whenever the mode of solution there adopted seems 

 to fail, the failure is apparent only, and the general solution may al- 

 ways be deduced from it." 



Several points of resemblance between the integral calculus and 

 that of functions, are then noticed ; and a remarkable analogy be- 

 tween a method of integrating differential equations, and a mode of 

 solving functional equations, is pointed out ; in both cases the ope- 

 rations are performed by multiplying by a factor, whose form is to 

 be determined by another equation. Some equations are given in 

 which this method is successful, and the obstacles to its general ap- 

 plication are pointed out as demanding further inquiry. 



Of the Construction of Logarithmic Tables. By Thomas Knight, Esq. 

 Communicated by Taylor Combe, Esq. Sec. R.S. Read February 

 27, 1817. [Phil. Trans. 1817, p. 217.] 



