484 ON CERTAIN APPROXIMATE FORMULAE FOR [59 



Now, from above, 



/(<) = < (sec <r>, 

 and therefore 



and = I -~ + (n+\) tan y. 



Hence 



= (sec y) m {l + -i- (a - /3) 2 |~2Zw f ^ ^ tan y + 2i (n + 1 ) (tan y) 2 

 1 24 v ' L \</>/o 



-i"j 



+ m [m + 1 (sec y) 2 m] 



= (sec y)'" 1 1 + (a - /3) 2 2lm (jrA tan y + m (TO + 2n + 3) (sec y)' J 



- m (m + 2n + 2) 1 1 . 

 Now make m + w+l = 2, or m= (n 1), and we have 



= (cosy)- jl -(a-/?) 2 [ 2 ? (-l) ( d ~ ) tan y +(n- l)(n + 4) (secy) 5 

 L ** \ ua *P/a 



In this make 1 = 2, and 1 = 1, successively, and we obtain the same 

 expressions for X and T as before. 



The case thus treated is not one of mere curiosity, but is practically 

 important. From theoretical considerations, Newton concluded that the 

 resistance of the air to the motion of projectiles is proportional to the 

 square of the velocity, and very little progress has been made in the 

 theory of the subject since his time. Experiments have shewn that the 

 relation between the velocity of a projectile and the resistance offered by 

 the air to its motion is far from being so simple as that given by 



