Vol. 8, 1922 
MATHEMATICS: J. L. WALSH 
139 
general parasitism, to a highly specific parasitism, in the latter cases the 
parasite having become so perfectly modified to fit itself to the one species 
of host that it must mean an advanced stage of evolution. 
In considering the importation of parasite species we must aim to have 
both kinds if they exist — the specifically constituted parasites and the gen- 
eral parasites. From the ranks of the latter we will have great help in 
the destruction of large numbers of the particular insect against which it 
was imported, but it will have the additional advantage of accommodating 
itself to other injurious species, and hence on the whole will probably be a 
more desirable addition to the American parasite fauna. 
Although not parasites, mention should be made of two species of ground- 
beetles brought over from Europe in the course of the gipsy moth work. 
Calosoma sycophanta has been found to attack a number of injurious 
native caterpillars; and Carabus auratus, while probably primarily a 
feeder upon slugs, as it is in its native home, also feeds upon cutworms and 
probably other soft-bodied insects ordinarily found in or near the ground. 
ON THE LOCATION OF THE ROOTS OF THE DERIVATIVE OF A 
POLYNOMIAL 
By J. L. WAI.SH 
Department of Mathematics, Harvard University 
Communicated April 11, 1922 
This note considers some geometric aspects of the problem of the ap- 
proximate location of the roots of the derivative f\z) of a polynomial 
f{z) when the roots of f{z) are known. If f{z) has Wi roots in a circle Ci, 
mi roots in a circle C2, and no other roots, the roots of f'{z) are known to 
lie in Ci, C2, and a certain third circle C which is readily determined. ^ 
Moreover, if f{z) has W] roots in Ci, W2 roots in C2, roots in C3, and no 
other roots, and if the circles Ci, C2, C3, are equal and have collinear centers, 
the roots oi f'{z) are known to lie in those circles and in two circles C, C" 
equal to the original circles and whose centers are collinear with their cen- 
ters.^ In each of these cases, the actual locus of the roots oif\z) consists of 
the circles stated, when the given circles are the loci of those roots of 7(0) 
(supposed to vary independently) which they contain. 
The second of these results suggests the problem of the determination 
of the locus of the roots of f'{z) when the locus of the roots oif{z) consists 
of three circles d, C2, C3 which are not supposed equal and with collinear 
centers.^ The solution of that problem is indicated in the present note; 
the answer is stated in the 
TheJor^m. Let circles Ci, C2, Cz he the respective loci of mi, W2, W3 roots 
of a polynomial f{z). Then the locus of the roots of f'{z) consists of Ci, C2, 
