266 mk de morgan, on a proof of the existence of 



p 



Whence — = cot {a^ + CTj + ... - ti - Tj - ...) 

 Q. 



Now it has been proved, algebraically, that for every one of the radical points, whether of 

 numerator or denominator, within the circuit, cr„ or t„ gains Stt ; on the circuit, ir continu- 

 ously, and TT per saltum without effect upon sign ; without the circuit, 0. The theorem is 

 now obvious. As to the excess of k over I, it matters nothing whether we make 6 pass from 

 to Stt, in any of the angles a^, a.^, ... tj, tj, ... consecutively, or in all at once. In the first 

 case, (Ti + (j'i + ... gives to the cotangent 2ot + m' changes of the form + — if the circuit be 

 convex, and none of the form - + : while if the circuit be not convex, the changes of the 

 first kind exceed those of the second by 2»» + m . At the same time, — tj — t^ — ... gives 

 an excess of — + changes over + — changes of 2 p + p. 



The theorem is universally true for all functions in which a root factor of the first dimen- 

 sion exists for every root. The proof most commonly given (the joint proof of Sturm and 

 Liouville) depends upon the consideration that where two closed circuits having no common 

 area have some portion of boundary circuit in common, the sum of the values o{ k — I for the 

 two separately is the value oi k - I for the single circuit made by neglecting the common 

 boundary. And this because the common boundary, being described in different directions in 

 the two circuits, contributes towards k in one circuit what it contributes towards I in the 

 other; and vice versa. Hence any circuit* may be divided into an infinite number of infi- 

 nitely small circuits ; and the theorem, being proved true for an infinitely small circuit, is 

 true for the circuit made of the outer line of all the subdivisions. There is no occasion, 

 after what precedes, to shew that if 



^(x + y ^ - 1) = {x + y '^ - 1 - a cos a - a sin «/;/- l)*'"\// (.1? + 2/\/ - ')> 



where \|/(acosa + a sina -y/- l), does not vanish, an infinitely small contour described about 

 the point {a cos a, a sin a) gives ^ - Z = ± 2»n or ± m, according as the point is within or upon 

 the contour. 



The theorem fails when the root factor enters with a fractional exponent : unless indeed 

 we propose an extension so vague as a theorem constructed on the trial of all integer powers 

 of ^s. 



Let the function be one in which every root-factor is of the first dimension, subject to the 

 usual definition of equal roots; and let it never become infinite for finite values of x and y. 

 Then the curves P = 0, Q = 0, the intersections of which determine the root-points, are such 

 that two branches, one of each curve, cannot inclose a space. At each root-point, the branches 

 which there intersect, must make known the existence of that root on every circuit which con- 

 tains the point, however large. The four places in which a branch of P = and one of Q = 

 meet any circuit, supposed convex, give -(- — , - « -i-, -h — , — oo -f, which are just suffi- 

 cient to indicate one root. No second root-point can then be determined by these branches 

 This is not, however, a definition of all curves which cannot inclose space ; for P = and 



• Those who remember the treatment of the electric circuit | ber that this is also the way in which an infinitely small cui- 

 by Ampere (I think, but it is long since I read it) will remem- ' rent is integrated into any current whatsoever. 



