ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 601 
It is thus only necessary to consider the second form. Look- 
ing to the essential simplifying condition of this form @ = 0, its 
integral has the following form :— 
a {cos (iu + ig! + A) du=aza, sin (iu +7'g' + A) 
+ aa;, sin ((¢+1)u+7g' + A) 
+aa;49sin ((¢@+2)u4ag' +A) + &e. 
+aa;_ sin (@—1)ut+i@g' +A) 
+aa;_ sin ((#—2)u+ig'+ A)+&c.; 
and it is shown in the memoir in question that the determination 
of the factors of integration «;, «;4 1, 4; 1, &c., depends on 
two rapidly converging continued fractions. These are,— 
oe 1 
oe; ea rae giv p I 
x a+2+i7y 1 
r iit Say i | 
——— — ke. 
A 
ie 1 
Bale @— 147 1 
rN co ee 1 
; o 
A ae, 
wherein, for brevity, there has been put 
= : eiy 
=F : 
When from these two continued fractions the numerical values 
a; a; 
of +" and — have been computed, and there has been 
a; a 
i i 
found 
then 
7) 
i Heli ap ng 
The same continued fractions serve to calculate the value of 
the ratio of any two consecutive factors of integration, and con- 
sequently these are all given. The integral 
a f sin ((u+ig' +A) du 
leads to an expression in which the factors of integration are the 
