500 Mr. H. Holt on a Method of Calculating 



Dividing each of these terms by c Xi 



k 

 0=3 — in the case when x — 1, 



-.(4F*+2cX) 



k 

 +.JT (B* + 2F, + D,) when c x =c,+c b 



Ox 



k 



+ §- (B* + 2F;-Dz) when c X =c x - Cli 



c x 



- 2(F*F m f 2cjG,F OT ) when p*= ^ + c m . . . (B) 



Adding this last equation to the equation (A), article 8, we 

 find, denoting B*-f2F, + D; by M,, and B ? + 2F;-D, by N,, 



(4 + P-l)F x =-pB, 



— 3k ( i + - ) in the case where x = 1, 



-F[i(M^-F z ) + ^M,] where c x + Cl =c x , 



-i*[i(Ni-F,) +-^ 3V *J where tfl -c,=c« 



+2 [(|F| + c,G )F W - i^G^ m G m ] where c* ± c m = c,. 



A: 

 6?^= | - in the case where x = \, 



c i 



-2F* 



/c 

 -f J - M; where c x + c t = c x , 



C x 



k 

 +|~ N ? where c l -ci=c x , 



"— 2(iF ? + cA)Fm where cj±£» '=<?,. 

 11. Hence we have the following Rules for calculating the In- 

 equalities of the second order : — 



*-'- d ^ .. ■ 



(1) Any inequality B z cos c^ in the value of -^ produces the 



Pi 

 following : 



Inequality in p = Fj cos c^, where Fj=— J £■ • *>/ 



»' ' - i >(A) 



Inequality in v= — -F^sin c//, where ^=£, 2 + p—l- j 



