34 Lord Kelvin on the 



The series (207) converges for all values of cr, great or small, 

 real or imaginary : (208) converges in its first i terms, if 

 2cr 2 >2? — 3 (modulus understood if o 2 is imaginary), and 

 after that it diverges, the true value being intermediate 

 between the sum of the convergent terms and this sum with 

 the first term of the divergent series added. The proper rule 1 

 of procedure to find the result with any desired degree of 

 accuracy, is to first calculate by the ultimately divergent 

 series, and see whether or not it gives the result accurately 

 enough. If it does not, use the convergent series (207), 

 which, by sufficient expenditure of arithmetical labour, will 

 certainly give the result with any degree of accuracy resolved 

 upon. 



§ 155. As a guide, not only for numerical calculation, but 

 for judging the character of the desired result without cal- 

 culation, it is convenient to find the moduluses of the three 

 complex arguments of the function E, in (205), and (206). 

 They are as follows : — 



mod (v'».*-^)=^/(0 + y +7) 



%\/f-, + < B \/-; wlien.r==x (209): 

 mod ( •«! + , ^-)= A /(g - ^ + ^) 



== t\ / -' - — co \/ ' - , when x = x (210) ; 



mod (^H\/' ( 2ii >- 



§ 156. The very interesting questions regarding the front 

 of the procession of waves in either direction, of which we 

 have found illustrations in figs. 36, 37, 38, aud which we had 

 under consideration in §§ 11-31, 114-117 above, are now 

 answerable in a thoroughly satisfactory mathematical manner, 

 by aid of the formulas (205), (206), (209), (210), (211). 

 When, in the arguments of E, in (205), and (206), ^/mt is 

 very great in comparison with Q)/2- v /y», the two added terms 

 in (205) are approximately equal, and (206) is reduced 

 approximately to its last term ; and all the solutions (189), 

 (190), become approximately sinusoidal, in respect to t. 



This is the case when t\/~r- is very great in comparison 



. . ' fx 



with unity, and in comparison with co\ / ~ , as we see by 



