272 Mr. E. Howard Smart on the Third-Order 



Also OP</G(P <+1 = («,+8»0/(*i+i + **+i)- • • (3) 



Hence (2) gives 



above expression for Q;Pi _ corresponding expression for QiP 1+ i ' 

 Let (4) be expanded by Taylor's theorem. Denoting 



we get 



* i + bXi — + l;ji ^— + As { s; r- ... 



0«?i O^ 0*i 



— t? .*,. ^Zi±l_i_^,i ^F t+1 , a,»S^»+i , 

 = ±i i+ 1 + d.tf f+1 ^- — + bf/i+i s- — + as { --j- + 



We develop this expansion in terms of 



(5) 



5 —j (KC, 

 Si 5,- S. 



su})posed of the same order of magnitude. 

 Fi (up to terms of the third order) 



Si \ *« Vi Sj + 5 ; 2 IV J> 



milking use of (1), and F l+ i=a similar expression with 

 Also to^lyJurLMjific-T} 



from the figure (;V) 



Eliminating &r., fy., && €+l , Sy i+1 with the help of these 

 from (5), we get, after a little reduction, as the coefficient of 



As; 



— on the left-hand side of (5), 



Si — Ti V " 



K-M&j + Vil/i+ £(* t — ty) -r A ) 



with a similar expression on the right-hand side. 



