558 A NEAV PBOOF OF THE PABAXLELOGBAM OF FOBCE8. 



tween zero and 2; and hence its value can always be represented by 

 2 cos (f>. But since it is desirable to take as little for granted as 

 possible, I stall demonstrate that R Y lies between the specified limits, 

 without availing myself of this additional axiom. Should R x not lie 

 between zero and 2, it may be denoted by x + x~ l , x being a pos- 

 itive quantity; and then R % =x* +x~* R m = x m + x~ m ; 



where all the quantities, R*, Rs, R , are positive. But this is 



impossible : for. — being . - — - ,it is plain that there is some number 



m, such that m- times —is intermediate between — and .in which 



2 2 2' 



case Rm-, the resultant of two equal forces, which act in the direc- 

 tions of the lines containing the angle rn6, is negative. Consequently 

 R, never can be > 2 ; and therefore we can assume 



R x = 2 cos <£. Hence also 



R a = 2 cos 2$ 



R^ = 2 cos m(f). 

 In proceeding to determine <£, we shall inquire after its least posi- 

 tive value, which must be < — . Now I say that the least multiple 



ft TT 



of— which exceeds sir + __, is the same with the least multiple of <f>, 



■_, — 



which exceeds «7r+— •. * being any whole number. For suppose that 



z 



the law holds as far as (s — 1) ^ + 1-. Represent by 



o 



the multiples of —between (s—l) tt + _ and sir + — Then since 

 2 2 2 



the quantities R^ , R/i+i, Rk , have all the same sign, the an- 

 gles h<f>, (h + 1) <£, .. (£— 1) <£, of which they are the double cosines, 



must be betwixt (Y— 1) w + — and s'it + — , *' being a whole num- 



2 2 



ber. But h<f> lies between (s— 1) ■* + '— and stt + — ; therefore, s'=s ; 



2 2 



and (k— 1) <£ agrees with (&— 1) — in being < s tt •+- —. But 



2. 2 



— > 9 ir + ~ ; and, since — - gives rise to a resultant of a 

 2 2- 2 



