70 
where 
ON THE INVERSION OE A QUADRIC SURFACE. 
[498 
, . a a 
= & + d’ 
An 1 & V 
~ ^ 1 ~~‘ 2 d i+ d } 
a 1 « , c 
- 47j - - \ + -j, 
— 4S X = 
— 8ai = — 
— 86j = — 
— 8c! = — 
d 2 d 
^ + ™?b' + n 2 c') + ~kj 4 > 
Hence Kummer’s equation 
Si + X 2 = 
4 d 
la 
d 2 ’ 
mb' 
~¥’ 
nc' 
d 2 ' 
-j- ) 
X + Uj X + fii X + >y l 
b i C! 
T 
or say 
becomes 
256a! 2 , 256&i 2 , 256ci 2 
64§! + 64X 4X + ^ + 4/3i + 4X + 4 7i > 
64X 2 — ^ (Pa 4- m-b' + n 2 c') — %. 
d 3 v 7 d i 
4>l 2 a 2 
< i ‘(*-5-a+ 4x ) *(*!-*+* 
+ 
4<m 2 b' 2 
a b' 
d 2 d ) V d 2 d J \ 2 d 2 d / 
+ 
4-nV 2 
a c' 
which is satisfied by 4X = — Writing therefore 
that is 
the equation is 
. a 6 
ix+2 d>-~d 
4 m 2 b' 2 
ir + 
4?i 2 c' 2 
4# 2 4#a 4 /; , ,, ,, 4l 2 a 2 
7, 4* —— j? (Pet + m 2 o + me) —^ yr + n —,, - r ^ > 
" •< - w-i-3 * -S-| 
79 / ,7/ , „ / /) ¿) l7 Pa' 2 m 2 b' 2 n 2 c' 2 
la + mJ + rac - 0a - M = W+i + 0Tb' + 0Tc'’ 
viz. this is