v! U:LOGRAM OP FOI: 



: ion </> (x) SB 2 COS . r = , v, 



shall next shew that if <f>(x) = 2 cos./ ,:?, ami 



7, and when x = /3-7, then ^>(x) = 2cosa; fl 

 IV m (1) 



= 4 cos ft cos 7 - 2 cos (y9 - 7) 

 = 2 cos ( + 7). 



-') holds when z = , it will hold wh.-u rr = 2; thi.s 



we obtain by supposing y = /3. : -' In -Ms wln-n x = /3 



and when a; = 2/9, it also holds when x = 3# ; and so on : 



is, if (2) holds when x = ft it will Imld wlu-n x = mfi. Tim* 



>nclude that if (2) holds when X = OL it will hold when 



X = , where m and n are any integ- 



]^ut since the numbers m and n may be as great as we 

 please, we can take them such that the expression 



differ as little as we pit-as;' from any assigned value of x. 



may therefore consider (2) as completely demons! 

 if it holds for any value of x dittVrmt iVnm /< r<>. Int by 

 Art. 16, it does hold when x= JTT, for then <f> (x) = 1 = 2cos JTT; 

 hence it holds always. Hence 



E - 2Pcos x. 



1: thru tlif forces Pbe represent, -d by strai.irht linf-s drawn 

 from their point of application, the resultant R will 

 sen ted by that diagonal of the parallelogram descril" -d mi 

 these straight lines which passes through the point of appli- 

 n. 



:t, let two unequal forces Pand Q act on the particle M 

 along the straight lines MA and MI! ; 

 represent tin-. the 



straight lines MG and Mil takm 

 ctions, and compl'-tn 

 the parallelogram J///A7/. 



First suppose AUB a right an- 

 gle. Draw the two diagonals MK 



