44.0 REPORT—1904. 
the parabola type : 
Ep eal AG 
@ —#)*” 
the curves given by 
Qotnta,U 4+ ...+a =0, 
where a, and a, _, are conjugate and one of these pairs varies with /. 
2. Peano’s Symbolic Method. By A. N. Wurrenzand, F.R.S, 
“3. The Theory of Linear Partial Differential Equations. 
By Major P. A. MacManoy, Se.D., FBS. 
4, On the Roots of the Characteristic Equation of a Linear Substitution. 
By T. J. Pa. Bromwicn. 
The equation in 2, 
| 4, iA, M4, 2 + + +3 Gyn =0, 
| Ga, 1) @o,a—-A, « + +) Aan 
i : - 8 / 
Gn, 19 En, 2 + + 43 Gy,n—Xr | 
or, let us say briefly, a—A =0, has been much discussed. It is known that the 
roots are real if @ is a symmetric matrix; pure imaginaries (or zero) if a@ is an 
alternate matrix ; and that the absolute value of all the roots is unity in case a 
represents an orthogonal substitution. 
If no information is available about special relations amongst the letters a, it 
is no longer possible to make such definite statements. The following theorem 
throws some light on the general case :— 
Write b,, .=3(y, s+ As, y)5 Cr, s=3(G;, s— as, ,) 80 that the matrix b is symmetric 
and ¢ 1s alternate ; let the roots of \b— =0 be a,, ay, ay, where a,>a,>... >ay; 
and let the roots of ec— =0 be + ty,, £7 «+ +9 E Mp, Wherey,>yo> +--+.» >yYp>O0. 
Then the veal part of any root of a~d =0 hes between a, and a,; and the absolute 
value of the imaginary part is not greater than y,. 
The first part of this theorem is due to Bendixson;' the second seems to be 
new. 
Hitherto all the letters a, , have been taken to be real; if they are complex a 
modification of the theorem is necessary. Let the accented letter a’,,, represent 
the conjugate complex to a,,,. Then write 
by c= Gy, gt Cs, 5), Ch e—a(Gr,s—Os »); 
so that the matrices 6, c have the property expressed by the equations 
hs SU ESS oe 
Tt is known that the two equations in A, 
\b-—rA =0, e~rA =0, 
have real roots in consequence of the property just stated: denote these roots by 
By, Boy « « «y Bn ANd yy Yoy « + +» Yn TeSpectively, and suppose them again arranged 
in order of magnitude. Then the equation a—\ =0 has the property that the real 
' Ofversigt af K. Vet. Akad. Férh. Stockholm, 1900; Acta Mathematica, t. 25, 
1902, p. 359, 
