33] 
Suppose, moreover, 
ON THE REDUCTION OF du + JU, &C. 
225 
we have evidently 
Hence writing 
and therefore 
we obtain 
f k 
= V/ 
~\P> 
I 
= ae 
— 4 hd 
+ 3c 2 , 
F 
= ae 
-m 
+ 3c' 2 , 
J 
= ace 
— ad 2 
— hé 2 — c 3 4- 2 bed, 
U' 
= a'c'e' 
- a'd' 2 
— Ve' 2 — c' 3 + 2b'c'd' ; 
xdy — 
II 
^"5 
k (x'dy' — y dx'), 
xdy — 
ydx 
j x'dy' — y'dx 
JP 
L JP' ■ 
u = y, 
w=K-, 
X 
x ’ 
xdy — ydx 
du 
x'dy' — y'dx' du' 
P^ 
~ U*’ 
►OH 
II 
»SH 
du 
7 du' 
7ü 
= k w 
the equation between u and u' being 
\ + a v! 
U = \ + ^u' 
Next, to determine the relations between the coefficients of U and 17'. Since P, P' 
are obtained from each other by linear transformations {Math. Journal, vol. iv. p. 208), 
[13, p. 94], we have between the coefficients of these functions and of the transforming 
equations, the relations 
P = fc I, 
J' = PJ- 
whence also 
J' 2 _ .J 2 
F 3 ” I 3 ’ 
Suppose now 
or 
whence also 
c. 
U' = a' (1 + pu' 3 ) (1 + qu'-), 
h' = 0, d' — 0, 6c' = a' (p + q), e’ = a'pq ; 
F = a' 2 (p 2 + q 2 + 14pq), 
J' = «' 3 (p + ?) ( S4> P2 -P 2 - J 2 ) ; 
29