434 
Proceedings of Poyal /Society of Edinburgh. [sess. 
(207), 
1.3.5 
(2a- 2 ) 8 + 
(208). 
The series (207) converges for all values of cr, great or small, real 
or imaginary : (208) converges in its first i terms, if 2o- 2 >2^'-3 
(modulus understood if cr 2 is imaginary), and after that it diverges, 
the true value being intermediate between the sum of the con- 
vergent terms and this sum with the first term of the divergent 
series added. The proper rule of procedure to find the result with 
any desired degree of accuracy, is to first calculate by the ulti- 
mately 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, hut for 
judging the character of the desired result without calculation, it 
is convenient to find the moduluses of the three complex arguments 
of the function E, in (205), and (206). They are as follows : — 
§ 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, and 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 
( 210 ); 
( 211 ). 
