114] STEINERS EXTENSION OF MALFATTl’s PROBLEM. 69 
It is obviously convenient that A'A" + B'B" should be symmetrical with respect to 
Y and Z, and this will be the case if 
+ @'y" = d'ft' + /3'y, 
that is, if /3' (7" - a") = /3" (7' - a') ; 
or assuming that the equations are symmetrically related to the system, we have the 
first set of relations between the coefficients, relations which are satisfied by 
a = 7 + 2cf)/3, a'= y' + 2cj)f3', a" = 7" + 2<^3 // , 
and the values of cl, a', cl" will be considered henceforth as given by these conditions. 
We have 
A'A" + B'B" - G'G" = a'a" + ¡3'f3" + (y'/3" + y"/3' + 2</>/3'/3") (F+ £) + (/373" + y'y") YZ 
+ -~{a + /3 (Y+Z) + yYZ). 
The quantities A' 2 + B' 2 — C' 2 , A" 2 + B" 2 — C" 2 are quadratic functions of Z and Y respectively, 
and with proper relations between the coefficients, we may assume 
(A' 2 + B’ 2 - C' 2 ) (A" 2 + B" 2 - G" 2 ) = l 2 s 2 {U 2 + k[(a + /3(Y + Z) + yYZf -8 2 (1 + Y 2 )(1 + Z 2 )]}, 
in which U is a linear function of Y + Z and YZ, and k and Is are constants. The 
first side must, in the first place, be symmetrical with respect to F and Z, or 
a! 2 + /3' 2 
-S'’-, (a' +/) /3', /3'=+ 7 '» 
must be proportional to 
a!' 2 + /3" 2 
-S"‘, (a" + 7") 8", ii"- + 7™ 
But since 
(*' + y)0, («" + 7")/8" 
are proportional to 
r 
1 
it is only necessary that 
¡3' 2 + y't _ S' 2 , (3" 2 + y" 2 - 8" 2 
should be proportional to 
r 
1 
or since the equations are supposed symmetrically related to the system, we must have 
the second set of relations between the coefficients. Suppose 
/32 + t 2 - S 2 /3 /2 + y 2 — S' 2 I3" 2 + y" 2 - B" 2 _ s 
« ~ <y'- _ oc' 2 y" 2 — 4 
y 2 — cC- — — 
then since 
4 (7 + cf>l3) 4>/3, &c.,