127] 
OF THE SECOND ORDER INTO ITSELF. 
135 
for the equations of the lines PA, QA respectively; and we have therefore the 
coordinates of the point A, coordinates which must satisfy the equation 
/3« + ay — 8z — yw = 0 
of the plane 6. This gives rise to the equation 
y 2 {ay x - 8z x ) - w 2 (yy x - fa) = 0. 
We have in like manner 
for the equations of the lines PB, QB respectively; and we may thence find the 
coordinates of the point B, coordinates which must satisfy the equation 
{3'x + a'y — 8'z — y'w = 0 
of the plane </>. This gives rise to the equation 
2/2 («Vi - 7 w i) “ *2 (%i - fai). 
It is easy, by means of these two equations and the equation x 2 y 2 — z 2 w 2 = 0, to form 
the system 
«2 = («2/i “ s *i) («Vi ~ l w i)> 
2/2 = (72/1 ~ ß z i) (%! “ ß' w i), 
¿2 = (72/1 - ß z i)( a 'yi ~ y' w i)> 
= («2/i “ ^1) (%! - ß'wß ; 
or, effecting the multiplications and replacing z 2 w x by x x y x , the values of x 2 , y 2 , z 2 , w t 
contain the common factor y x , which may be rejected. Also introducing on the left- 
hand sides the common factor MM', where M 2 = a/3 — 7S, M' 2 = a'j3' — 7'S', the equations 
become 
MM'x 2 = y'Sx x + 0La'y x — a'Sz x — ay 'w x , 
MM'y 2 = ßß'xj + yS'y-j - ßS'z x - ß'yw x , 
MM'z 2 — ßyx x + ya'y x — ßa'z 1 — 77 'w x , 
MM'w 2 = ß'8x x + aS'y x — 88'z x — aß'w x , 
values which give identically x 2 y 2 —z 2 w 2 = x x y x —z x w x . Moreover, by forming the value 
of the determinant, it is easy to verify that the transformation is in fact an im 
proper one. We have thus obtained the equations for the improper transformation of 
the surface xy — zw = 0 into itself. By writing x x + iy x , x x — iy 1 for x x , y x , &c., we have 
the following system of equations, in which (a, b, c, d), (a', b', c', d') represent, as 
before, the coordinates of the poles of the plane sections, and M 2 = a 2 + b 2 + c 2 + d 2 , 
M' 2 = a' 2 + b' 2 + c' 2 + d' 2 , viz. the system 1 
1 The system is very similar in form to, but is essentially different from, that which could be obtained 
from the theory of quaternions by writing 
MM' (?« 2 + ix 2 +jy 2 + kz 2 ) = (d + ia +jb + kc) (iv + ix +jy + kz) (d' + ia! +jb' + kc'); 
the last-mentioned transformation is, in fact, proper, and not improper.