59] CALCULATING THE TRAJECTORIES OF SHOT. 477 



Since ^ = -w" +l (seci)' l+l ) 



d(f> g 



this last expression may be put under the form 



Also J F(y) = M '(secy) m . 



Hence, by the above lemma, 



(" u l (sec <f>) m d ( f ) = (a- ft) u t l (sec y) m |l + ^ (a - ft)"- 1"? (I + n) (-^rV 



+ l(2m+n + l)(- 7) tany + m(m + l)(secy) 2 -m"ll 



' \ud<f>/ JJ 



where ( rr] denotes what T-T- becomes when a> 0, or when y is sub- 

 \udAL ud<t> ' 



stituted for (f>, and for M, that is 



/ du \ ^ 



The factor u l may be eliminated from this expression, and the expression 

 itself simplified, by means of the formula 



1 1 A [" 1 du - k(n-l) f a ,, 



n^i n~i = \n 1} I , , -j--. cl<p w (sec <p) dq>. 



q p J P u "9 # J /3 



for, putting m = n + 1 in the above expression, we have 



f 1 f 



u l (sec <^>)" +1 d(f> = (a ft) u ' (sec y)" + i 1 + (a /3) 2 Z (Z + n) 



tany + n+ 1 w + 2 (sec y) 2 - (n + l)*l I . 



JJ 



Hence ' sec > m d<)^ u l sec 



' (sec ^>) m d<j)^ \u l (s 

 J/3 



I 1 +7rr( a -) 2 r^(w-n-l) f J^-) tany + m-n- 1 m + n+~2 (secy) 

 I 24 ' \ud<p/ t 



m w 



