IN A RESISTING MEDIUM. 
133 
Mr. Bashforth puts n — 3, and then 
p= 3 r\\+ P *)dp 
Jv 
— 3a + a s — 3p — p 3 , . 
.(5)* 
4 __ w {P a dp 
(J J P ( 3 a + d 6 — 3p —p d )i 3 
.( 6 )* 
m _ vP p a dp 
gjp (3a + a s — 3p — p 3 )% 3 
. (7)* 
p a pjp 
y ffjp (3a + a s-3p-ff . 
.( 8 )* 
Here 3a 4* a s = -, u Q = wy*; and Mr. Bashforth has calculated by 
quadratures the values of these integrals for different values of y, and 
tabulated them in Tables IY. and YI. 
But the substitution P — (a — p) s z s reduces these integrals to elliptic 
integrals; for, putting a — p — - , then 
and therefore 
3 (1 + « 2 ) - 3aq + 1 = * 3 , 
aq a A 
+ 
>3 _ 1 
/2 
+ 
2 3 — P 
and 
putting 
Therefore P~% dp 
I+a? 1 4(1 + a 3 ) 2 3(H-« 2 ) ' 4(1 + a 2 ) 3 3 + 3« 2 ’ 
a . V(2 3 -^) 
9 2 + 2a 2 + ^(3 + 3^)’ 
4 + « 2 
4 + 4a 3 
= b\ 
dp 
(a — p ) 2 z* 
_<k = U 3 \ 
z 2 v \4 + 4« 2 / 
dz 
V (> 3 - 6 3 ) 
dz 
V (z S — & 3 ) ’ 
and therefore % = 
an elliptic integral of the first kind. 
Also, ^ = n/( 3 + SaY-JYr 
therefore pP'Up = (a - 1) 
(9) 
a 
z s — 'l 
dz 
_ 3 ] 
dz 
2 0 3 -l } 
* Bashforth. c< Motion of Projectiles,” p. 53. 
