447] 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
235 
104. To effect the foregoing transformation, writing 
x' : y' : z : w' = {x — py){ xw — yz) 
: {x- y)(qxw-pyz) 
: (z — qw) ( xw — yz) 
: (z — w) (qxw — pyz) ; 
or what will ultimately be the same thing, but it is more convenient for working with,, 
x’ = (x-py)( xw- yz), 
y' = {as- y) {qxw -pyz), 
z = (,z — qw){ xw— yz), 
w' = (z — w) {qxw — pyz); 
these give 
x — py = M'x, 
x — y — N'y', 
z — qw — M'z', 
z — w = N'w’, 
where M', N' are quantities which have to be determined; and thence 
(1 - p) x = M'x — pN'y, 
(1 — p) y — M'x' — N'y, 
(1 — q) z = M'z' — qN'w', 
(1 — q)w — M'z' — N'w'; 
whence also 
(1 - p) (1 — q) ( xw- yz) = N' [ {{q - 1) x'w' - (p - 1 )y'z’} M' + (p - q) y'w'N'], 
(1 - p) (1 — q) {qxw — pyz) = M' [- (p - q) x'z'M' + {{pq - q) x'w' - (pq —p) yz'} iT] ; 
but we have 
or, substituting, 
that is 
xw — yz x x—py_x _ M'x' _N\ 
qxw — pyz y' ' x — y y' N'y' M' ’ 
M' { (q - 1) x'w' -(p- 1) y'z'} +N'(p- q) y'w' 
= M' {- (p - q) x'z'} + N {{pq - q) x'w' - {pq -p) y'z'}; 
M' {{q - 1) x'w' -{p- 1) y'z' + {p-q) x'z'} = N' {{pq - q) x'w' - {pq - p) y'z' - {p - q) y'w'}; 
or, what is the same thing, 
M' = {pq - q) %'w' - (pq - p) y' z ' - (p - q) y'w', 
N' = { q- 1) x'w' -{ p- 1) y'z' + {p-q) x'z'; 
30—2