398 
PEOFESSOK CAYLEY ON THE TEANSFOEMATION 
for greater clearness in a square form, and adding to them those for the non-prime cases 
n= 9 and n — 15 ; also a valuable table given by him for the powers of f(q) ; and I give 
other tabular results which are of assistance in the theory. 
The General Problem . — Article Nos. 1 to 6. 
1. Taking n a given odd number, I write 
1—?/ 1-ar /P-Q*\ 2 
\-¥y~\+x \T-fQa?J ’ 
where P, Q are rational and integral functions of x 2 , P + Qx being each of them of the 
order — 1), or, what is the same thing, (1+#)(P + Q#) 2 being each of them of the 
order n ; that is, 
n=4p-\- 1, n= 42^+3, 
Order of P in x 2 is p , p, 
„• Q „ p— 1, p; 
whence in the first case No. of coefficients of P and Q is (j? + l)+p, = and 
in the second case No. is =i(w-j-l), as before. Taking 
P = a+yx 2 -\-sx 4 + . . . , 
Q=j3 + ^ 2 -f £r 4 + 
the formula is 
1 —y \—x /«— / 'ix + yx 2 — . . .\ 2 
1 +y~ 1 +x \a + / 3x + yx 2 + . . ’ 
the number of coefficients being as just explained. Starting herefrom I reproduce in a 
somewhat altered form the investigation in the ‘ Fundamenta Nova,’ as follows. 
2. If the coefficients are such that the equation remains true when we therein change 
simultaneously x into — and y into — , then the variables x, y will satisfy the differential 
equation 
M dy 
dx 
^ l - f. I — K 2 y- V\ —x^ . 1 — £ V s 
1 2/3 \ 
(M a constant, the value of which, as will appear, is given by u=l+ : ^- J ; and the 
problem of transformation is thus to find the coefficients so that the equation may 
remain true on the above simultaneous change of the values of x, y. 
In fact, observing that the original equation and therefore the new equation are each 
satisfied on changing therein simultaneously x, y into — x , — y, it follows that the equation 
may be written in the four forms 
i -y =(i-*) A2 (-)> i+y =(i+®) B2 (-) 5 
1 — Ay = (1 — Jcx)C*(-~ ), l+Xy=(l+/hdC 2 (>), 
the common denominator being, say E, where A, B, C, D, E are all of them rational 
and integral functions of x ; and this being so, the differential equation will be satisfied. 
