132 CELESTIAL MECHANICS. I. H. 11. 



tion is somewhat less than 45^, and sometimes, when the 

 velocity is very great, as little as 30^. But it usually 

 happens in the operations of natural causes, that near the 

 point where any quantity is greatest or least, its variation 

 is slower than elsewhere : a small difference, therefore, in 

 the angle of elevation, is of little consequence to the ex- 

 tent of the range, provided that it continue between the 

 limits of 45^^ and 35^ ; and for the same reason, the angular 

 adjustment requires less accuracy in this position than in 

 any other, which, besides the economy of powder, makes 

 it in all respects the best elevation for practice, where the 

 object is to carry a ball or shell to the greatest possible 

 distance.] 



[277. Lemma A. If the equation a + hx 

 •\-cx^ + dx^^ . . . = be true for all values of 

 a?5 it will follow that each coefficient must be 

 separately =0, 



For, putting rr zzO, we have azzd, therefore 5x -f cx^ + ... 

 =0; then, dividing by ar, h -\- ex ■\- dx^ -{- ., . =0; conse- 

 quently 6=1:0 ; and in the same manner all the coefficients 

 may be made to vanish in succession. 



278. Lemma B. The binomial or dino- 

 mial theorem (244) is true for all powers, 

 whether entire or fractional. 



Its truth may be the most easily shown from the principles 



of fluxions, and the Taylorian theorem combined. For 



since d(a;'0=:wx"-idx/ making dx constant, we have di\x^) 



= w(w — 1) a;''-2 dx2, and d3(x") =/i («— 1) in— 2) x"-^ d^^^ . 



, , dw' h^ 

 whence, taking uzz (x-fAy% we have Aw =^"5 — '"Ts 



