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947] AND OTHER SYSTEMS OF TWO TETRADS. 
We interchange herein e, f and also E, Z, and we thus obtain 
A'+CE + B'Z 4- D'EZ = 0, 
where 
A' = . (/3 — 8) a7e + (7 — a) /38£ 4- (a8 — (3y) ef, 
=-(7-a)^S-(a^-7S)e . + {/3-8)e£ 
G = — (¡3 — 8) ay . 4- (a/3 — 78) £ + (7 — a) e£, 
E' = - (a8 - /3y) - (7 - a) e -((3-8)1; 
viz. this is the condition for the existence of the axis d 23 i l3 i 13 i i3 i i3 . 
11. I remark that we have 
A' = A + a (e - £), B' = B + b (e - £)> 
C' = C+ c(e-C), i)' = Z>+d( e -£), 
where 
a = (¡3 - 8) ay - (7 - a) /38 = a(3 (7 + 8) - 78 (a + /3), 
b = c = 7 8 — a/3, 
d = a+ (3 — 7 — 8, 
and further that 
B-C=a/3(y + 8)-y8(a + /3) + (y8- a/3) (e + f) + (a + /3 - 7 - 8) e£ 
say this is II. 
12. It thus appears that, for the determination of E, Z, we have 
A + BE +CZ + DEZ = 0, 
A' + C'E + B'Z + D'EZ= 0. 
Eliminating Z, we find 
A + BE _ G + DE 
A' + C'E~ B' + D'E' 
that is, 
(AG' — A'C) + (AU - A'D + BE - GC) E + (BE - ED) E* = 0; 
upon reducing the coefficients of this equation it appears that they contain each of 
them the factor II, and throwing out this factor, the equation is 
e [a/3 (7 - 8) + 78 (/3 — a) + (a8 — (3y) f] 
+ [— a/3 (7 — 8) — 78 (/3 — a) + (aS — (3y) e + ((3y — a8) £+ (ft + 7 — a — 8) ef] E 
4- [(3y — a8 — (/3 + 7 — a — 5)] E- = 0; 
this contains obviously the factor E—e, or throwing out this factor, we have for E 
the simple equation 
[a/3 (7 — 8) + 78 (/3 — a)} 4- («8 — /3y) (E 4- £) 4- (/3 — a 4- 7 — 8) %E = 0. 
In a similar manner it may be shown that the two equations give for Z the like 
simple equation 
{a/3 (7 — 8) 4- 78 (/3 — a)} + (aS - /3y) (Z 4- e) + ((3 — a 4- 7 — 8) eZ = 0, 
viz. starting from the chords 1, 2, 3 which depend on the parameters (e, £), (a, /3), 
(7, 8) respectively, these last two equations give the parameters (E, Z) of the chord 4.