A BOOT IN EVERY ALGEBRAIC EQUATION. 263 



for a contour which is only part of a circuit to be changed into part of another contour having 

 a different value of k - I, without passing over one or more of what I call radical points. 



If (^iz be a rational and integral function, the possibility of assigning closed circuits which 

 have different values oi k — Ih easily shewn. When an angle gains a revolution by continued 

 increase, the cotangent of that angle passes through two changes of the form + — , and two 

 of the form — oo +. When the gain of a revolution is a balance of increase and diminution, 

 every case of — + which occurs during diminution is accompanied by a case of + — which 

 occurs during restoration, over and above the two cases of + — which belong to the balance. 



P . 



Consequently, whenever — is the cotangent of an angle which gains a revolution during the 



Q. 



progress of {x, y) round a closed circuit, k — I acquires two units in that revolution, and two 



units in every such revolution. If the angle change only by increase, we have k •= 2, I = 0, 



for each revolution. 



Representing a (cos a + sin a ^- 1) by a^, &c., and ce + yxZ—loTr (cos + sin ^- 1) 

 by Vg, let ^ (r + y \/- 1) be a^Vg" + b^Vg"'^ + ... +m^: in which, to avoid a visibly existing 

 root, we suppose that m has value. We see then that 



P ar" cos(n9 + a) + 6r''~'cos[(n - 1)9 + /3] + ... + mcos/ti 

 Q ar" sin {nO + a) + br"''^ sin [(« - 1) 9 + j3] + ... + m sin ft.' 



If a closed circuit be taken in which all the values of r are infinitely small, we see that 

 P : Q is either constant, or, where cos fi or sin ix vanishes, varies directly or inversely as the 

 cosine or sine of a multiple of 9 altered by a constant. In these cases each revolution gives k 

 and I both = 0, or both the same integer : that is, k — I = 0. But if throughout the closed 

 circuit r be infinitely great, the value of JP : Q is always cot {n9 + a) and k — I acquires two 

 units for each accession of Zir which n9 + a receives, while 9 changes from to Stt : that is, 

 k - I = 2n. Hence the proposition that (pas always has a root or roots is proved. We then, 

 in the common way, establish the existence of the root-factor, and the number of the roots. 



Previously to proceeding further, I discuss a point which is of great importance, and bears 

 on many of the proofs of the preceding proposition. Dr Peacock {Report on Analysis, 

 p. 305) objects to making interpretation the foundation of important symbolical truths, 

 which, he maintains, should be considered as necessary results of the first principles of algebra, 

 and ought to admit of demonstration by the aid of those principles alone. 



Interpretation is, or at least begins with, the application of meaning of fundamental 

 symbols to the deduction of meaning for compound symbols. It may be applied to throw 

 light on the steps of a demonstration, and in this way it must be applied : without it algebra 

 is a valley of dry bones. It may also be applied to furnish steps of demonstration ; and this 

 sort of application must be sternly resisted : the result is not algebra. But on this point the 

 following distinction suggests itself. 



Every proposition is true of which the truth can be shewn. Demonstration of the possi- 

 bility of demonstration is itself demonstration ; demonstration of the possibility of demonstrat- 

 ing the possibility of demonstration is also demonstration : and so on. Mathematical teaching 

 has used this principle rather too extensively. A proposition proved to be true of commen- 

 surables is allowed to be assumed as to incommensurables, on the feeling that its truth as to 



