OF ARBITRARY CONSTANTS, &c. 



125 



APPENDIX. 



QAdded since the reading of the Paper.] 



On account of the strange appearance of figures 2 and 3, the reader may be pleased to 

 see a numerical verification of the discontinuity which has been shewn to exist in the values 

 of the arbitrary constants. I subjoin therefore the numerical calculation of the integral to 

 which fig. 2 relates, for two values of x, from the ascending and descending series separately. 

 For this integral D = 0, and I will take C = 1, which gives, (equations 30,) 



^ = 7r-^r(l); B= -S7r-5r(f); 

 and log J = 0-1793878 ; log (- B) = 0-3602028. 



The two values of x chosen for calculation have 2 for their common modulus, and 90«, 

 1.60°, respectively, for their amplitudes, so that the corresponding radii in fig. 2 are situated 

 at 30" on each side of the radius passing through the point of discontinuity c. The terms of 

 the descending series are calculated to 7 places of decimals. As the modulus of the result 

 has afterwards to be multiplied by a number exceeding 40, it is needless to retain more than 

 6 decimal places in the ascending series. In the multiplications required after summation, 

 7-figure logarithms were employed. The results are given to 7 significant figures, that is, to 

 5 places of decimals. 



The following is the calculation by ascending series for the amplitude 90" of a?. By 

 the first and second series are meant respectively those which have A, B for their coefficients 

 in equation (19). 



