522 Sir William Thomson on Stationary 



To find now the effect of a single ridge, remark that if I be 

 the length from ridge to ridge, 



m = 27r/l (19). 



After the manner of Fourier now suppose I infinitely large ; 

 which makes m infinitely small ; and put 



im = q and m = dq ..... (20) ; 



then with *=1, (15) becomes 



2A/7T . cos qx , 01 N 



=j • • • i*-u ; 



where 



h=W!g (22). 



Equation (21) will be shortened, and for some interpretations 

 simplified, by making ^D = cr, when it becomes 



m I n 



Jo 6 <r_j_ 6 -<r__ — ( 6 <r__ 



M _ . . (23) , 



bo ^-'^ 



The definite integral (21) or (23) seemed rather intractable, 

 and the quadratures required to evaluate it, for many and wide- 

 spread enough values of x to show the shape of the surface 

 for any one particular value of D/b, would be very laborious. 

 But I had found a method of evaluating it from the periodic 

 solution for an endless succession of equidistant equal ridges 

 (15), wholly analogous to analytical deductions from corre- 

 sponding solutions for cases of thermal conduction and of 

 signalling through submarine cables, to be found in vol. ii. 

 pp. 49 and 56 of my Collected Mathematical and Physical 

 Papers ; and, towards applying this method to a particular 

 case of the disturbance due to a single ridge, I had fully 

 worked out the periodic solution for the case represented by 

 the diagram of curves (fig. 3, p. 529), when I found a direct 

 and complete analytical solution for the single-ridge problem 

 in a form exceedingly convenient for arithmetical computation, 

 except for the case of x equal to zero, or from zero to a quarter 

 or a half of the depth. The previous method happily gives 

 the solution for small values of a, and indeed for values up to 

 two or three times the depth, by very rapidly converging 

 series, and thus between the two methods we have a remark- 

 ably satisfactory solution of the whole problem. 



Before explaining the curves and their relation to the pro- 

 blem of the single ridge, I shall give the new direct solution 



