306 On the Parallelogram of Forces. 



with the direction of a?, which equals 29 ; then z evidently bisects the 

 angle 20, formed by the directions of R and S, it also equals the sum 

 of their components which act in its direction; hence and by (1), we 



have z =<p(3), & a?=(R + S) . <p(6) 2 =zp(6)— the resultant re- 



R-f-S 



solved in the direction of x ; by resolving S in the direction of x, 

 and adding R which acts in that direction, we have R+S.p(23) 

 the components resolved in the direction of x 9 which must equal the 

 resultant resolved in the same direction ; hence we shall have 

 (R+S).9(^=R+S ? (24),or since R=S we shall have 2 ? («) s 



l+(p(2t)) 5 (3); which must evidently be an identical equation. 



It is manifest by (I), that if 4 = 0, <p(0) and <p(2d) will each =1 ; 

 hence supposing <p(d) to be converted into a series, arranged accord- 

 ing to the ascending powers of 0, its first term must = 1, and the 

 powers of 6 must be positive, for should any of them be negative, 

 <p(d) would be infinite when 4 = 0, instead of being =1, as we have 

 proved it must be j .". <p(d) must be of the form, <p(d) = l+A0 a + 

 BJ 6 +Cc) c +DJ d +,&,c. (4), and by changing 6 into 24, we shall 

 have the expression for <p(2d) ; by substituting the values of 9(4), <p(24) 

 in (3), we have 2(1+A4 a +, &c.) 2 = 2+2 a . A4 a + , &c., or 4Ad a 

 + 2(A 2 4 2a + 2BJ 6 ) + 4(AB3<"+*> + Cd c ) + 2(B 2 6* b + 2Dd d + 

 2AO a+c) )+, &c.=2 a . A4«+2*. Bl b +2*. Q c +2 d . D4<*+, kc. 



(5). Since (5) is to be identical, (so that 6 may be indeterminate,) 



it is evident that the coefficients of 4 a , must be equal ; .\ 4=2°, 



which gives a =2, but A remains undetermined ; by substituting the 



value of a, and comparing the next higher powers of 4, we have 



2(A 2 d*+2Bj»)=2 6 BJ 6 , which requires that 6=4, .\ A 2 +2B 



A 2 

 8B, or B = __ ; in the same way we find c=6, d=8, &c. C 



A3 ^4 



, D = — , and so on. It is evident that 0(6) must gen- 



2.3 2 .5. 2.3M.5.7 rv ' 



erally be less than 1, .\ put A= - — , and we have B=— — —j C 



2 JfmO»*T 



., D = , and so on, where the law of contin- 



2.3.4.5.6 2.3.4. 5.6.7. S 



uation is manifest; hence by substituting the values of a, J, &c, A, 



B, fee., m (4), we have <p(0) = 1 — + — — +,&c. 



' w w 2 2.3.4 2.3.4.5.6 



which is the well known expression for cos. k$; hence <p(0)=cos.W> 

 which substituted in (1) gives, x=*z cos. Jc9. To determine Jc, let 



