22 
ON A DIFFERENTIAL EQUATION. 
[710 
where P, Q are quadric functions of x; P', Q' quadric functions of z. But P and 
Q may contain a common factor, and the integral is then expressible in the form 
P' 
x = q , the quotient of two quadric functions of z; or P’ and Q' may have a common 
P 
factor, and the integral is then expressible in the form z—^, the quotient of two 
quadric functions of x; or there may be a common factor of P, Q, and also a common 
factor of P' and Q', and the integral is then of the form 2 — jj^> the quotient of two 
linear functions of x. 
In the general case the differential equation is 
A (aP' + bQ') dz 2 (aP + bQ) dx 2 
z 2 (z-l) 2 
x 2 (x— l) 2 
where a, b are arbitrary constants, X is a constant the value of which can in each 
P 
particular case be at once determined; so when the integral is z= ^, the differential 
equation is 
A (az + b) dz 2 _ (ciP + bQ) dx 2 
z 2 (z—l) 2 x 2 (x—l) 2 
where a, b are arbitrary constants, but A is now a linear function of 2 the value 
of which can in each particular case be at once determined. When the integral 
is z = the differential equation is 
A (az 2 + 2 bz + c) dz 2 _ (ciL 2 + 2b LM + cilf 2 ) dx 2 
z 2 (z — l) 2 x 2 (x — l) 2 
containing the three arbitrary constants a, b, c; A is a constant the value of which can 
be at once determined. 
There are in all 6 integrals of the form z = jj-> f° r which the differential equation 
P 
F I 
contains three arbitrary constants: 18 integrals of the form z = -q ^and of course the 
P' 
P P' 
same number of integrals of the form x = -q,j, and 9 integrals of the form f° r 
of which the differential equation contains two arbitrary constants. It is to be remarked 
that Rummer, considering the values of z as a function of x, obtains the 72 rational and 
irrational values mentioned in his equations (31), (35), (36), (37), (38), and (39): but the 
72 values are made up as follows, viz. the 18 values of z as a rational function of x, the 
36 irrational values obtained from the 18 expressions of x as a rational function of and 
the 18 irrational values of z obtained from the 9 integrals in which neither of the 
variables is a rational function of the other: 18 + 36 + 18 = 72.