STEINER’S EXTENSION OF MALFATTl’S PROBLEM. 
263 
Writing' the second and third equations under the form 
A'+B'X+CVT+X^=0 
A"+B"X+CVT+^=0, 
the result of the elimination may be presented in the form 
A'A" + B'B" - C'C" = v/A'^+B'^-C'Va"^+B"^-C'^ 
which is most easily obtained by writing X= tan (p and operating with the symbol 
COS"* ; but if the rationalized equations be represented by 
X'+2ijJX+u'X^=0 and X"+2|t^"X+/'X^=0, 
the form 
4 (}!v —pJ^) = {X'v” 
leads easily to the result in question. The values which enter are 
A' = a' + f3'Z A"=a"+i3"Y 
B'=3'+7'Z B"=i3"+/Y 
c'=sVr+z^ c"=§Vr+^; 
whence, in the first place, by the equation connecting Y, Z, 
8 ' 8 " 
C'C"=-y-{a+^(Y+Z)+WZ}. 
It is obviously convenient that A'A"d-B'B" should be symmetrical with respect to 
Y and Z, and this will be the case if 
u'^''+(By = a"(B'+(^y, i. e. if i3'(7"-a")=|3"(/-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=yd-2<p(3, a'=y'd-2<?>|3', a" =y" -\-2<pj5'' , 
and the values of a!, a', a" will be considered henceforth as given by these conditions. 
We have ^ 
A'A''+B'B''-C'C''=aV'+/3'^3'' + (yi3''+/^'+2(p/3*/3'')(Y+Z) + (|3'(3''+y'/)YZ 
8 ' 8 " 
+-^{«+^(Y+Z)+yYZ}. 
The quantities A'^d-B'*—C'^, A"^+B"*— C" are quadratic functions of Z and Y respect- 
ively, and with proper relations between the coefficients, we may assume 
(A'^_j_B'^-C'^)(A"^-fB"^-C"^) = /V{U**+^[(a+|3(Y+Z)+yYZ)^-S^(l-l-Y^)(l+Z^)]}, 
in which U is a linear function of Y-f-Z and YZ, and k and Is are constants. The 
first side must, in the first place, be symmetrical Moth respect to Y and Z, or 
(a'+yjiS', 
must be proportional to 
2 M 
MDCCCLII, 
