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Cz {except that Ci is to he omitted if mi = i) and of a region R bounded by the 
bicircular quartic 
m,H{x-h)^ + {y-hy-r2''] [{x-c,y + (y-c,)^-r,^] 
+min{x-a,)^ + (y-a2)2-ri2] [(x-c^y + (y-c,y-r,^] x 
+m5^[{x-ai)^ + (y-a2)^-ri^ [{x-hy + (y-b2y-r2'] 
(1) +2m2m3[(x-ai)2 + (y-a2)'-n^ [(x-h) (x-c,) + (y-b2( (>' - C2) - ^2^3 ] 
+2mmz[(x-bi)^ + (y-h^-r^^ [(x-ay) (x-c,) + (y-a2) {y-C2)-r,rs] 
2mm2[{x-cxy + {y-czY-rz^] [{x-ai){x-h{) + (t-qs) (3'-&2)-n^2] = 0; 
/j^r^ the circles Ci, C2, C3 have the respective centers and radii: {ai,a2),r^; 
(61, 62), r^; {ci, C2), f3. If the curve (i) consists of two nested ovals, R is the 
region hounded hy the outer oval; if (i) does not consist of two nested ovals, R 
is the interior of the curve. 
If no two of the regions C\, Ci, Cz, R have any point in common, they con- 
tain respectively mi — i, mi — i, mz — 1,2 roots of f'{z). If R is in two 
parts, which have no point in common with each other or any of the regions Ci 
which is a part of the locus of the roots of the derivative, each part of R con- 
tains one root of f'{z). 
This theorem replaces the problem of the exact location of the roots of 
f'{z), which depends on the solution of an equation of degree wi + W2 + 
mz — 1, by the problem of their approximation which is solved by plot- 
ting a quartic curve, which depends on the solution of equations of degree 4. 
In particular cases — such as that mentioned in the first paragraph of 
this note — the quartic (1) degenerates, in which case the new problem is 
still simpler, for it depends merely on the solution of quadratics. 
The theorem can be proved by a straightforward method which we now 
indicate. The locus of the roots of f'{z) is readily found when C\ is not 
a null-circle but C2 and Cz are both null-circles ; every point of the boundary 
is a point of the curve (1) where f2 = ^3 = 0. The locus of the roots of 
f'{z) can then be found when neither d nor C2 is a null-circle by deter- 
mining the envelope of the former boundary; every point of the new 
boundary is a point of (1) where fs = 0. Finally the locus is found in 
the general case by determining the envelope of this last determined bound- 
ary ; every point of the boundary is a point of the curve (1). 
A particularly interesting case of the theorem occurs if Wi = m2 = mz, 
ri = f2 = rzy and the centers of Ci, Ci, Cz are the vertices of an equilateral 
triangle. The locus of the roots of f'{z) is (in addition to the regions Q) 
the interior and boundary of a circle whose center is the center of this tri- 
angle and whose radius is {r\ -f rih)'^', h being the distance from the 
center of the triangle to the centers of the circles Q. 
The geometry of this entire situation deserves further inquiry, with es- 
pecial reference to the cases where (1) degenerates. Our theorem general- 
izes presumably to the case in which any number of circles Q form the 
