474 ON CERTAIN APPROXIMATE FORMULA FOR [59 



Hence the velocity at time t is u sec <f), and we may denote the 

 resistance by ku n (sec <j>) n , where k is constant throughout the small arc in 

 question. 



Also let p and q denote the values of u at the beginning and end 

 of the arc, a and ft the corresponding values of <j), g the force of gravity, 

 T the time taken to describe the arc, X and Y the corresponding total 

 horizontal and vertical motion. 



Making < the independent variable, the fundamental formulae are 



du ku n+t 





, dx U* 



< 2 > d$ = ~g 



(3) ^=-| 



(4) fj= -^ 



d(j) g 



From the first of these equations 



1 du k , ,, + 



= ~ ( sec vF ' 



^ 



-s+7 ~n ~ 

 u , d(f) g 



and therefore, by integration between the limits (f> = a and <f> = ft, 



r/ p g 



Also, we have 



1 f a 



X = - u 1 (sec (fry d<f> ; 

 9 J ft 



I f 

 = - 



9 J ft 



and 



T=- 



and we wish to compare the two former of these definite integrals with 

 the following known one, viz. : 



1 1 , . f" 1 du ,, k(n-2) f a 

 ^r a -- irr 2 =(w-2) z=i-TrdAs*-* 



n p n ' ) pu n l d$ g J p 



