270 



Mr. W. D. Niven on the Calculation 



t, the time ; 



x, the horizontal distance from some fixed point ; 

 y, the vertical distance. 



The integrations will be performed over a component arc of the trajec- 

 tory, and the three last quantities are measured from the beginning of 

 the arc. The values of t, x, and y over the whole arc will be denoted by 

 T, X, and Y. The values of u at the beginning and end of the arc will 

 be denoted byp and q; those of by a and /3. The acceleration due to 

 gravity is denoted by g. 



Piest Method. 



§ 3. The solution adopted by Mr. Bashforth is the famous one first 

 given by John Bernoulli and published in 1721. It applies to any 

 retardation formulated by fiv n . All the characteristic quantities are 

 expressed in terms of 0, which may be accomplished briefly thus : — 



The equations of motion in the horizontal direction and in the direc- 

 tion of the normal to the trajectory in the ascending branch are 



pv n cos (1) 



« ^ = -9 eos (2) 



Hence 



^t=JVH-* (3) 



city g 



9 



Integrate this, recollecting that the initial values of u and (j> are p and 

 a, and get 



1_1 



— A = Jt I n sec' !+1 d> dd>. 



If the symbol P<p denotes I n sec" +1 <p cty, the equation will become 

 Jo 



u n jp M g K 97 

 The horizontal velocity at the end of the arc is therefore given by 



I=I + ^(P«-P P ) (A) 



2 P 9 



