438 THE THEOBY OF SCREWS. [400
It follows that D A and (B + C) A must both be independent of 8. We
may therefore make
A = A'-, B = B'+S; C = C'-A- D=D‘-
and thus the homographic equation becomes
A Xp + (2? + A) X + ((7 — A) p + D' = 0,
where A is the only quantity which involves 8.
The equation can receive a much simpler form by taking the infinite
objects as the two originating objects from which the range was determined.
In this case the equation
■4X2 + (B + O) X + D = 0
must have as roots X = 0 and X = oo, and therefore
.4=0; 2) = 0,
hence the homographic equation reduces to
BX + Cp = 0;
since B 4- C is a function of the intervene 8, we may say, conversely, that
We have, however, already learned that the intervene is to have the
form
</> O) - </> (/*)•
Now we find that it can also be expressed, with perfect generality, in the
form
It follows that these two expressions must be equal, so that
£ (X)-<£(/*) = ^Q).
In this equation the particular value of 5 does not appear, nor is it even
implied. The formula must therefore represent an identical result true for
all values of X and all values of p.
Wo may, therefore, differentiate the formula with respect both to X and
to p, and thus we obtain