116 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Using (63) in the fourth form of (58) we have 
e a—b z 
io Fa - (65) 
and using (64) we have 
an \* 
a= (ya) 
The nome of the complementary integral is denoted by Jacobi 
and writers that follow him by 7. In our system this would be 
the notation for the nome of the integral ¢’/; ¢/’ that for gy”, ete; 
also gq, that of (¢,)y1.3 q% of (%)y, ete. It is therefore better to 
follow the example of Broch, who denotes the nome of the com- 
plementary integral by p. We have then 
aay 
pik POE ae 25] 
Se OOt) BO ae oth) eee aro T= Pa (sr) (67) 
where aml) = 7/ Gi) g @—1) 
a (7—2) — V at—qnr2 
a (n—3) — V an—2 qn—3 
al! = V a” q 
a =VY aa (68) 
By (55) and the second forms of (58) and (67) we have the 
following relation between p and q 
ip-4lq-4—= PP (69) 
or in Briggian logarithms 
log {log p— log q~4} = log (_| log e)? = 9.6678084 (69) 
or _—_ log{log p-! log q-! } = 0.2698684 (70) 
By means of this relation we can always choose the shortest 
route to either p or g. It is easy to see that the nomes and com- 
