OUR COMMON ROADS—MURPHY. 461 
the perpendicular and 6 the base of a right angle triangle, of 
which the hypothenuse B C represents the slope of the angle of 
repose : 
which somewhat exceeds the greatest admissable gradient. For 
simplicity the load may be supposed to rest on a single wheel, 
shewn in the figure. In the smallest triangle T is the perpendic- 
ular, P the base, and W the hypothenuse in which 
Py a 
From the two similar triangles 
t h 
tiprvh:boe—=>— 
IP b 
by substitution 
t . 
pee eee rc) h 
yr p= ; but & 
being the perpendicular divided by the base, represents the angle 
at the base or the angle of repose, and this is the maxi- 
mum admissable gradient. Hence the gradient should not exceed 
the quotient obtained by dividing the force of traction by the 
square root of the difference between the square of the load and 
the square of the traction. Upon good roads ¢ is so very small 
in proportion to W that it may be omitted in the denominator, 
and we have practically for the angle of repose~, or the force of 
traction divided by the weight of vehicles and load. 
For No. 4 the formula becomes. 
65 
or 1 perpendicular to 34 base, and generally the proper grade 
for any kind of road, or the ratio of the vertical to the hori- 
