104 FIFTH REPORT—1835. 
Involving, and arranging according to the powers of q , gives 
y = a (f — ( 2 a p — 2 a w — tv) q 2 — (2 p w x 1 + « — & jp 2 
— 2 a/ 2 ) q -h Wjp* — 2 ™ maximum. 
7 
.*. ^ = 3 « — 2 (2 ay — 2 e zc — ?/;) ^ — (2 p w x H« 
— « p 2 — 2 ?c 2 ) = 0; 
d2 y 
d q 2 
= 6 « 
— 2' (2 a p — 2 a w — w). 
Since = 0, 
d q 
.3 a cp — 2{2 ap —2 aw — w) q — 2pw x 1 4 - a — a p~ — 2 w l . 
From this quadratic equation we obtain 
If we substitute this value of q in that 
we have 
r / 2 // 
d <f 
2 p iv 
i 9 
+ w 
The negative and affirmative signs preceding the radical here, 
show that in the two values of which q admits, the positive 
radical answers to a minimum, and the negative to the maxi¬ 
mum required. 
If the beam be uniform, r — q (Cor. 2. Prob. 2.), and a = 1. 
In that case, the maximum impact gives, by equation (A), 
2 i _ 
q — x p — w — — \/p 2 + 3 iv 2 . . . . (B.) 
Cor. If iv be very small compared with p , then 3 w 2 is so 
compared with p 2 . Neglecting 3 tv°~, in the last equation, gives 
the value of q , when the resistance to impact is a maximum, 
1 I 
= — p — iv, nearly; or q -f iv = — p 9 nearly. 
This corollary, and especially the more general values of q in 
the problem, would enable us to adjust the weight of a bridge 
