PARTICULARLY THOSE OF TWO VARIABLES. 
d /& 
ifak,/A A_„. A ^ 
C809 
— C 4< C 5 GO C 6 C 7 0 Q 
12/ YL2 "12 " 13! " 
12 "12 
cc AfAU-v A A. 
/ ^3_ ^7_ 
12 ^12 
which correspond to Bosenhain’s (104), (105). Let 
^ 1 ==a! i(l —— K i 2 a 5 i)(l “^iK 1 “ K s%) 
X 2 = %(1 —Xo)(l— k l 2 x 2 )(1 — K%X 2 )(l “ 
805 
(55) 
(56) 
(57) 
Substituting in (53), (54) from (52), 
efe 2 , «{x/X^fl—/c/%) — %%,)} —fi{ ^/X^l — w 2 % 2 ) — \/X**!(1 — *A.)} 
Xl tic +X *di— 
(l- £c] )'t+( 1 ^) 
d.% 
cfe 
«{v'X 1 (l-ai 8 )(l-^)-V'X,(l- ah )(l-^)}-/3{v'X 1 (l- a ! 8 )(l-« !i %)-'/X 1 (l-» 1 )(l-.-/ J !i)} 
Wry 
where a, /3 are functions of k l , k 2 , k 3 and of c 5 , c 7 which will be afterwards seen to be 
themselves functions of #q, k 2 , k 3 . From these 
dx 2 _/3(1 — k^x x ) — «(1 — /Cg 2 ^) /_ 7 + 
, - say; 
rlw fY* — -T 3 ^ /yi - 'Y* A ' v 
vvtv tC-'o ^Oc> tUi 
f&q a(l — /c 3 2 * 3 ) —/3(1 — /c 2 2 5Co) 
clx 
Similarly 
If, then, we write 
v/X 1= - 
<&r 3 y' -f S , x 1 y^r- 
dy~ : r “' v/ J 
7 + &j /«- 
1^2 
Vx, 
a? 3 —a?]_ 
dy x 2 —x 1 1 ‘ 
A + Baq 7 , A + Ba ? 2 
1 
^ V5f rf35i+ Vxf d * 2 
, A' + B g, . I 
2/_ ®* 1+ ® 2 J 
(58) 
the foregoing equations are satisfied, provided 
