346 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
This expression may be reduced as follows : 
Let P=W, Q = (a^-&^)[/c^+a"-6^], (122.) 
and the preceding equation will become 
O=/d0^P+Q sin®&+R siiP0 (123.) 
Let this trinomial be put under the form of a product of two quadratic factors, 
(A+B sin^6)(C— B sin^0)=AC+B(C— A) sin^0— B^ sin^0 (124.) 
Comparing this expression with the preceding in (121.), we get 
AC=6^P, C-k=¥ya^-h\ B=a^-¥ (125.) 
A + B 
To integrate (123.) : assume tan^<p= — ^^tan^6 (126.) 
The limits of integration of the complete functions will continue as before. Making 
the substitutions indicated by the preceding transformations, the integral will now 
become 
'■d?>[l-§(|^)sinV] 
a/C(A + B) ^v_| 
AC 
1 
B 
A + B 
sitff] \/ l-§(CT)s'nV 
A + B— C (A + B)“* 
Q =-, N=l— wsin^, 1= 1 — sin^ip, 
and the preceding expression may be written 
(128.) 
[2n — — 
(.on, 
5(1^)*“' 'Jnwi’ ^ ' 
V ny — n)[\ 
It will presently be shown that A and C must always have the same sign, whence r>«. 
1 + 
A 
As and as C is always greater than B, ^^< 1. From (125.) we may derive 
(A + B)(C-B) 52 AC 
F— (C-A-B)2 ’ ^2 — (c_A-B)2' 
Now, that the values of a and h may be real, we must have C>B, while A and C 
must be of the same sign ; but as B is essentially positive, C, and therefore A, must 
be positive. 
O- 18 . A + C , 
Since A+B~”’ ~' C (128.) 
we may eliminate A, B, C from the values of the semiaxes of the base of the elliptic 
cylinder, and express a, b and /r, in terms of i and n. We may thus obtain 
\2n-?-ny ’ ( 13 ^-; 
In order that these values of a and h may be real, we must have n positive, i^>n, 
and l>i^. 
This is Case VI. in the Table, p. 316. 
