1916-17.] Adelphic Integral of Differential Equations. 
103 
solved for <p :i : and thus combining our results we obtain lor our integral- 
series (p 
Constant = <p=s 1 q 1 - s 2 q 2 + ^(Uj cosp 1 + U 2 cos 3 p } ) 
+ M2 4 { - U 3 C0S P 2 + o" 1 ~ cos ( 2 ih +P- 2 ) + cos ( 2 Pi -P 2 ) j 
+ 9r T ? 2 { U 6 GOS Pi + COS (2 p 2 + p 1 ) + cos (2 p 2 -i?d} 
+ - U 9 cos^> 2 - U 10 cos 3 j> 2 J 
{sTi^W + ♦*•}-*♦ U, + X "}“* 4 "‘ 
6s x S 2 
+ 2i 
+ r, % » I C0S (Pi +Pi) / _ 2s l TT TT 
+? i 2 ! |— r^--- 1 • -> 1 ^ 
o w 7 -T— r— ; — r UrUo XJolJo 
Sj + 2^2 u 7 (2s 1 - s 2 )(s 1 - 2 s 2 ) 5 8 3 6 
+ 
4s 
6 Sr 
\ 
2s 
^U 4 U 6 - + ^-^U 2 U 5 + (s, - s 2 )X 4 
q 1 “j" ^9 09 J 
C OS (Pj p 2 ) f _ 6s ± S 2 TT TT 
Si-s 2 t (2 s 1 + s 2 )(s 1 ' x 4 7 
S 1 ~ '^ S 2 
fc-U 8 U 8 - U 3 U 6 
/ 1 .s ^ 
+ 
2s 
A-U 5 U 6 - UjU 3 - - 2l ^- U 2 U 4 + (s 1 + s 2 )X 5 ) 
+ 
cos (3 p x + j» 2 ) f 1 0 g i g 2 u U + S - 2 - 
3sj + s 2 - ^ 2 X ^1 4" 2s 2 ) 7 2s t + Sr 
2s 2 
U 4 U 6 - 3U 2 U 3 
2s x -f- s, 
■E x U 4 + (3s x - s 2 )X ( 
+ 
COS (3j9 4 -p 2 ) f 10s 4 s s 
ILIL 
3s 4 - s 2 l (2 s 4 + s 2 )(s 4 - 2s 2 ) ^ 4 ^ 8 ' 9 
Si - s 2 
U 5 U 6 - 3U 2 U 3 
2 s, 
+ ?1?S 
u 4 u 9 + u,u a + 
cos 2 », / — — _ _ _ „ . , 
' 1 1 2s 4 + So 2s'i — Sr 
2s x - s. 
Bso 
5^9 
Uibg + (3s x + s 2 )X 7 
) 
U„U 
2 (^1 4 ~ ^Soji^s-^ 2 s 2 ) 
^-4— U 3 U 4 - n — " — U,U, + X„| 
2si + s 2 3 4 2s x -s 2 5 9 J 
9 9 
+ cos 2 H^y - ^y° + yy + yy° 
+ 
4- 
cos (2pi + 2 p 2 ) j _^f]_U 4 U 9 + _Jh_UJJ 
hi 1 s' 7 
2si + 2s 2 12si + s 2 4 9 2si-s 2 5 10 s 4 + 2s 2 
* ^ft» u ‘ D ’ + A n - u - - *k v ' v - *' i ° h - 2 '- ,x ") 
+ ^ pPi-y ,) f - JVu.F, 
2s x -|- So 
2s 1 - 2s 2 l 2 s 4 - s 2 6 9 ' r 
4s 
+ -y~vy 
s i 2s 2 
6^8 
A DiD> - £k v ' v ’ - Srt nA ! ‘= ,x ”) 
