570 
Proceedings of Royal Society of Edinburgh. [sess. 
§44. Return to our dynamical solution (75) ; and remark that 
if J is an integer, one term of (75) is infinite, of which the 
dynamical meaning is clear in (70). Hence to have every term 
of (75) finite we must have J =j + 8, where j is an integer and 8 is 
<1 ; and we may conveniently write (75) as follows : 
cl = c 
<!+ + + 
ecos 0 e 2 cos 20 
+ ~ cv . + 
or 
S+j-l S+j - 2 
e j+1 cos (j + 1)0 e j+ ' 2 cos (j + 2)0 
1-8 2-8 
d= c^+ J 
+ 
e? cos jO 
ad 
inf. | 
( 82 ); 
(83), 
where $ and J denote finite and infinite series shown in (82). 
§ 45. We are going to make 8 = |+ and in this case J can be 
summed, in finite terms, as follows. First multiply each term by 
gi+s e~i- s ; and we find 
-g cos (j + 1 )6 + cos (j +2)9 + etc. 
/= -c(8+j)e>+ s 
= - c(S+j)ei +s J deye~ 8 cos(j + 1)0 + 6 1- 5 cos (j + 2)0 + etc. 
= - c(8 +j)ei t 5 j" de e _5 {RS}^+ 1 (l + eg + e 2 <f + etc.) ; 
where q denotes e ld ; and, as in § 3 above, {RS} denotes 
realisation by taking half sum for ± i. Summing the infinite 
series, and performing Jde , for the case 8 = J, we find 
J= - c(j + |)e?+*{RS}g?+* log 1 + +! 
1 -Jqe 
■ (84), 
- + + 4)^+*{ks}^ +i i<>g l 
v 2/ 1 J 1 - Je cos 40 - ije sin A 
= kcO' + 4)e»'+4{RS}g/+ 
where 
ij/ — tan - 1 _ + sin ^ 
and therefore 
0 
Je cos \0 - ije sin ^0 
\ / 1 +2 Je cos 40 + 6 , , . .,s 
log V + t(l/f “ ^ 
i ]/ = tan 
-i , . . (85), 
+ Je cos ^0 1 T l-^/ecosJ 
— tan -1 sin ^ 
