318 Proceedings of Royal Society of Edinburgh. [sess. 
§ 25. The function D(:r, t\ which appears in § 13 as an item in 
one of the modes of summing shown for P(a:, 0) in (19'), and 
indicated for T?(x, t) at the end of § 13, and which has been used 
in some of our summations for Q(a;, t) ; is represented in figs. 6 and 
7, for t = 0, and t = 25r respectively. 
§ 26. Except for a few of the points of fig. 6, representing 
D(£, 0), the calculation has been performed solely for integral 
values of x. It seemed at first scarcely to be expected that a fair 
graphic representation could be drawn from so few calculated points; 
but the curves have actually been drawn by Mr Witherington with 
no other knowledge than these points, except information as to all 
zeros (curve cutting the line of abscissas), through the whole 
range of each curve. The calculated points are marked on each 
curve : and it seems certain that, with the knowledge of the zeros, 
the true curve must lie very close in each case to that drawn by 
Mr Witherington. 
§ 27. The calculation of Q(:e, t), for positive integral values of x, 
is greatly eased by the following arrangements for avoiding the 
necessity for direct summation of a sluggishly convergent infinite 
series shown in (46), by use of our knowledge of T(x, t). We 
have, by (46) and (19), 
Q(0, t) = 10(0, t) - 0(1, t) + 0(2, t) - ad. inf. (48), 
P(0,0-2f(-l)W<) • • • • ( 49 )- 
i= — oo 
Hence, in virtue of </>( -i,t) = t), 
P(0,<f§2Q(0,<) (50). 
Again going back to (46), we have 
Q(&, t) = 7£<h(x, t) - cf>(x + 1, t) + cf>(x + 2 , 2) - <ji(x + 3, t) + 
Q(a;+ 1,2)8 %<f>(x + 1, t) - <j>(x+ 2, t) + <f>(x+ 3, t) - 
By adding these we find 
Q(* + 1, t) + Q(x, t) = ^[4>(x, t - </>(£ + 1, 2)] = -|-D(£f, t) (51 ). 
By successive applications of this equation, we find 
'2Q(x + i,t) = (- l) < 2Q(a; > '2) 1 - ( - 1 ) < D(aj, t) ± . . + T>(x + % - 1, t) (52). 
Hence by putting = and using (50), we find finally 
2Q{*, 2) = ( - imo, 2) - ( - iyD(0, t) ± .. +D(*1,2) (53). 
This is thoroughly convenient to calculate Q(l, 2), Q(2, t) . . . . 
successively ; for plotting the points shown in fig. 9. 
