1887.] Theta Functions as Definite Integrals. 131 
§ 3. Similarly, by applying the fourth method described in 
§ 7 of the previous paper, to the q-series for © (w) and ©, (w), it 
is found that 
® (u) = 2, <p tet 
TL pp 
cosh mt — cos sim (2 at) dt 
=142{ A+B-—C-D 
0 
cosh art — cos wt 
cos 4 pt) dt, 
l 
“A—B+C=—D 
©) 9 cosh at + cos wt 
2| 
0 
cosh wt + cos wt 
sin (1 ut?) dt 
cos ($t’) dt*, 
where A =sinh (47 +2) tcos(47—2)t, 
B=sinh ($7 — x) tcos (474+ 2)t, 
C=cosh (47+) tsin (47 —2) ft, 
D=cosh (47 —«#)¢sin (47 + 2) t. 
In these formule x must not exceed the limits + 47. 
The functions @,(u) and ©,(w) are represented by exactly the 
same expressions as ©, (w) and © (uw) respectively, if we now sup- 
pose A, B, C, D to have the values 
A = sinh (7 — x) tcos at, 
B=sinh at cos (a — 2) t, 
C= cosh (7 — 2) t sin at, 
D= cosh at sin (a — @) t. 
These formule for ©,(w) and ©,(w) hold good when « does not 
exceed the limits 0 and z. 
§ 4. Since ©,(0) = Jp, © (0) = Jk’p, 8,(0) = Jhp, and 
©, (ox) = Jkk’p’« when «# is very small, we find by putting 
* Tn these formule for © (wu) and 0, (uw) we may express the four numerators 
A+B+0+D, A+B-C-—D, &ce. also in the forms 
2(P+Q), 2(P-Q), 2(P’-Q’), 2 (P’+Q’) 
respectively, where 
P=sinh $rt cos4rt cosh wt cos ct+cosh 47rt sin $rt sinh xt sin wt, 
Q=cosh 4rt sin 4rt cosh xt cos et — sinh drt cos $rt sinh t sin zt, 
P’=sinh 4rt sin drt cosh xt sin xt+cosh 4rt cos 47 sinh zt cos zt, 
Q’=cosh $rt cos4zt cosh at sin #t— sinh 4rt sin drt sinh xt cos zt, 
