A ROOT IN EVERY ALGEBRAIC EQUATION. 265 



We see that tan \^ makes the change + - or — + only when sin (9 - /u.) = 0, 

 cos (0 — |u) = =t 1. If r <m at 6 = n (which answers to taking the second point outside the 

 contour) tan \j/ in both cases goes through the changes of sin (9 - m) inverted : that is, through 

 + - and - +, at 6 = n, 9 = /j. + -tt. That is, tan xj/ returns to its first value, at = Zir. 

 by a balance of positive and negative revolutions : \|/ could not give a revolution without two 



— 0+ changes in the tangent. If r > m when 9= fx (which answers to taking the second point 

 inside the circuit), then, at 6 = /x, tan \jy goes through - +, and at 9 = fx + ir, also through 



- + : that is, tan >|/ recovers its first value at 9 = Zir by gaining a whole revolution. But if 



r = m when 9 = fi (which answers to taking a point on the contour) then tan \^ passes through - 



without change of sign, and s sin cr, or m(sin 9 — sin/u) changes sign without s changing sign: 

 that is, o- receives, per saltuoi, an accession or diminution of ir. 



At 9 = n + TT, tan \|/ undergoes the change — + which, not being, compensated until 

 9 = fj. + Stt, shews half a revolution added to yp by the time >// gains its original value. 



Appealing to the above as algebraical proof of the requisite property of the circuit, and 

 using the geometrical phrases only as combined abbreviation and elucidation, I shall now pro- 

 ceed to Cauchy's theorem, which with its extension is as follows. 



Let (pz be any rational function whatsoever, and (p{x + y \/~ l) being P + Q^y- 1, 



P 



let — be recorded while the point (*,y) describes any closed circuit in the positive (or rather 



Q 



positive-balance) direction of revolution. Let k be the number of + — changes, I the number 

 of — + changes. Let m and m' be the numbers of points within and upon the circuit, at 

 which (p{/v + y <^ — 1) = 0. Let p and p' be the numbers of points within and upon* the 

 circuit at which (p{x+y^— l) = o:. Then 



k — 1= 2m + m' — (2p + p). 

 Cauchy included only the case in which, by hypothesis, m' = 0, p = 0, p' = 0. 



Let the function be a rational algebraical fraction, in which the roots of the numerator 

 come under a„ (cos a„ + sin a„\/ — 1) and of the denominator under 6„ (cos j3„ + sin l3„\/—i). 

 Let the function he (p (x + y -y/ - 1), a? + y -y/ - 1 being r cos 6 + r sin 9\/ - 1, and let 

 r cos 9 - a„ cosa„ + (r sin - «„ sin a„) . \/ - 1 = s„ (cos ct„ + sin a„ y/ - 1), 

 r cos - 6„ cos /3„ + (r sin - 6„ cos /3„) . \/ - 1 = t^ (cos t„ + sin t„ -y/ - l). 

 The function <p{x + y y/ — 1) is therefore a constant multiplied by the following fraction 



«,S8 (cos £7] + sincr, -y/ — 1) (coscr, + sin o-o -^ — 1) ... 



t\ti (cosTj + sin Ti-y/— 1) (cosT2 + sin t^'^/ — I) ... 



♦ Sturm says, positively, that there can be no theorem when 

 a root is on the contour, for that different contours containing 

 the same numbers of radical points, may in that case give dif- 

 ferent values of k - I. But this was said after the first proof, 

 which he and Liouville gave together, and before the second 

 proof, which I am now translating into my own language, as 

 applied to the extended proposition. Had he reconsidered his 

 assertion while employed on the second proof, he could not have 



missed the introduction of m'. Any one who will take up the 

 point as a question of continuity by the aid of the curves 

 P = 0, Q = 0, will easily detect the loss of a change of the form 

 + — , or a gain of - +, when the circuit passes over an inter- 

 section of the curves P = 0, Q=0. In this he will need the 

 following theorem, which is easily proved : — when the circuit 

 passes through an intersection of P = t), Q = 0, either both P 

 and Q change sign, or neither. 



Vol. X. Part I. 34 



