262 Mr DE morgan, ON A PROOF OF THE EXISTENCE OF 



any alteration, unless by and <» coming together, whether with change of place or simple 

 recession. It is supposed that the series both begins and ends with a sign + or — , which 

 remains unaltered ; not with or oo . 



Except appulse of and oo , the only other changes are appearance or disappearance of 

 between like signs, appearance or disappearance of oo between like signs, opening of or 

 00 into or 00 oo with signs between them. A simple induction will shew that, in every 

 case which involves no appulse of and <» , either k and I remain unaltered, or receive the 

 same increment. 



Thus when + + changes into + +, both are unaltered : as also in - + changed into 

 -0-0 +, or-0- changed into - [ - -] — . But in + + changed into + [o - O] +, 

 both k and I increase by a unit : in + - changed into + [- + -] -, both receive a 

 unit of increase. But when - - is changed into -0[-oo +]0-, in which case k aug- 

 ments by a unit, while I is unchanged, the change, if continuous, commenced by an appulse of 

 and 00 , as in 00 0. Again, when — + co + changes through — Ox + to - oo + +, in 

 which case I loses a unit, there is an appulse of and oo , This theorem brings the funda- 

 mental theorem on the roots of equations to rest on what will readily be acknowledged to be 

 its proper foundation, the necessity of or oo in the transition from positive to negative. 



Now suppose a line of any sort drawn in a plane, and at each point of it, {x, y), let the 

 sign of a given function of a? and y be recorded ; with the character of each change, + -, 

 — +, + CO — , — 00 .+, as the case may be. Every contour, and every portion of a contour, 

 will thus present what we may call a chain of signs, such as+0— 0+ oo — 0+..., with re- 

 ference to any function of x and y which may be chosen. If the contour, or part of a 

 contour, change continuously, so as to pass gradually from one form and position to another, 

 changes may occur in the chain ; and it is obvious that the change may be so conducted, 

 that not more than one of the signs and oo shall be affected at any one moment. If the 



p 



function examined be — , where P and Q never become infinite for any finite values of ,t and 



y 



y, then can only appear when P = 0, and oo can only appear when Q = 0, and an appulse 



P 



of and 00 can only take place where — takes the form -. Next, suppose (px to be a 



function which never becomes infinite for any finite value of z, and let (pi^s + y \/— 1) = 

 P + Q\/— !• We see then that if A; — ^ be found to have, on one contour, a value different 

 from what it has on any other contour, a gradual transition from one contour to the other can- 

 not be made without the varying contour passing through points at which P — 0, Q = 0, or 

 f{z) = 0. Such point or points then must exist ; or we have the following 



Theorem. If f(x + y -y/- 1) = P + Q -^- 1, and if neither P nor Q can be infinite 

 for any finite values of on and y ; if also two contours can be found for which k — I has 

 different values ; then such difference of value is proof of the existence of a root or roots 

 which satisfy (pz = 0. 



It is supposed that the choice begins and ends with fixed signs. This always takes place 

 when we go round the whole of a closed circuit, from one sign to the same again. But we 

 have also seen that, so long as the initial and terminal signs remain the same, it is impossible 



