1893.] of circles ; and other systems of two tetrads. 59 



11. I remark that we have 



G'^G + c{e-0, D' = D + d(e-0> 

 where a =^(^-8) ay— (7 — a) /3S = a^ (7 + S) — 7S(a +/3X 

 b = c = 78 — cf/3 



d= a + /3 — 7 — S, 



and further that 



B-C=^a^(y + 8)-j8(a + /3) + (ryS-a^)(6+^) 



+ (a + /3-y-8)e^, 

 say this is 11. 



12. It thus appears that for the determination of E, Z wo 

 have 



A +BE + CZ +DEZ = 0, 

 A'+ C'E+B'Z+ D'EZ^O. 

 Eliminating Z we find 



A + BE C + BE 



.+ 



A'-v G'E" B' + D'E' 

 that is 

 (AB' - A'G) + {AD' - A'D + BB' - GC) E + {BU - B'D) E' = 0- 



upon reducing the coefficients of this equation it appears that they 

 contain each of them the factor 11, and throwing out this factor, 

 the equation is 



6 [a/S (7 - S) + 78 (a9 - a) + (a8 - /37) ^] 



+ [- a/3 (7 - 8) - 7S (/3 ^ a) + {aS - ^y) e 



+ (/37 - aS) ^+ (/3 + 7 - a - S) e^] E 



+ [/S7 - a8 - (/3 + 7 - a - 8) ^' = ; 



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)| + (a8 - ,87) {E + f ) 



+ (/3-a + 7^8)^^^ = 0, 



and in a similar manner it may be shown that the two equations 

 give for Z the like simple equation 



{a/3 (7 - 8) + 78 (/3 - a)] + {ah - /3y) {Z + e) 



+ {/3-a + y-8)eZ-- 0, 



viz. starting from the chords 1, 2, 3 which depend on the para- 

 meters (e, 5, (a, /3), (7, 8) respectively, these last two equations 

 give the parameters {E, Z) of the chord 4. 



5—2 



