402 Proceedings of Royal Society of Edinburgh, [sess. 1905 - 6 .] 
over 0 represents (137), for t = 0. The other curve of fig. 1, 
with positive and negative ordinates on the two sides of 0 , 
represents (137), with — {RD} instead of {RS}. The symbols 
{RS} and {RD} were introduced in § 3 above; {RS} to denote 
a realisation by taking half the sum of what is written after it 
with ±i, and {RD} to denote a realisation by taking — of the 
2c 
formula written after it minus — of the same formula with + 1 
Jii 
changed into - t . A new curve in which the ordinates are numeri- 
cally equal to fj2~dx ^he or( ^ na t es °f the second of the old 
curves of fig. 1, is now given in the accompanying diagram, fig. 33 ; 
and close above it the first of the old curves of fig. 1 is reproduced, 
with ordinates reduced in the ratio 2^/2 to 1 , for the sake of 
comparison with the new curve. This new curve represents the 
more convenient initiational form referred to in the title of the 
present paper. 
Its equation, found by taking t = 0 in (139) or in (144) [most 
easily from the imaginary form of (139)], is as follows : 
0 ) = 
1 
'2J2 
V(p+^) (2z - ) 
p 3 
. . (138). 
§ 101. The original derivation of the new particular solution, 
(which we shall call \j/,) from the primary (136), as indicated in 
§ 100, is shown by the following formula : 
, M) = { RI) } 
d_ -1 
dx + ix) 
-g# 
e 4(z+ta;) 
\ 
-g&z 
€ 4p2 
(139), 
where p = J(z 2 + x 2 ), and x = tan \x/z ) . 
An equivalent formula for the same derivation, which will be 
found more convenient in §§ 135—157 below, is as follows : 
^(£,M) = {RS} 
1 d 2 -1 
g dt 2 J(z + lx) 
- gt 2 
e 4(z+ia;) = 
-1 df 
gJ2 dt 2 
$(x,z, t) 
(140). 
