THE EOYAL ARTILLERY INSTITUTION. 
417 
a ring tire, and having its lower spoke vertical, is proportional to the sine of 
the angle of hollow, the strength of such wheel being dependent on the 
amount of resistance to extension exerted by the tire. 
Thus, if P represent the pressure or blow applied at <?, and tending to 
bend or break the spoke b at the point o } and S be the normal resultant of 
the resistance to extension opposed at c by the tire, then 
P = 8 sin a ; 
a being the angle of hollow. 
Further, we must bear in mind that the forces giving rise to this normal 
resistance in the tire actually produce a longitudinal extension of it propor¬ 
tional to its length, which, again, is proportional to the diameter of the 
wheel. Accordingly, the effective resistance of the tire, and therefore the 
strength of the wheel, is also inversely proportional to its height. 
If, then, 8 y and S represent the strength of two wheels of different dia¬ 
meters—for instance, the front and hind wheels of a wagon—having the 
same sectional dimensions of tire, the hollows of their arms and diameters 
being respectively a x and a, and h y and li> we have 
8 X : S :: 
sin a, sin a 
h x h 
and therefore if two such wheels are to possess the same strength, 
or 
sill a. 
sm a x 
sin a 
sm a 
h * 
V 
. *. sin a x = sin a. 
fb 
This value of a y shows the amount of hollow which should be given to 
the axletree-arms of the fore-carriage in order that its wheels should be 
equally strong with the wheels running behind them. 
We have already ascertained that the grind-pressure on the axletree-arms 
is proportional to the tangent of their hollow. Hence, when the tire of the 
front wheel must needs be kept as broad as that of the hind wheel, every 
constructive principle demands that the hollow of the arm should not exceed 
the amount imposed by the conditions of strength. These conditions we 
now see are satisfied if the sines of the hollows are proportional to the 
heights of their respective wheels. 
Thus in the 3 ft. 4 in. and 4 ft. 8 in. wheels of the service, the hollows of 
the arms should be as five to seven, at the outside; and, indeed, the propor¬ 
tion might be still further reduced with perfect safety and much benefit to 
the draught. 
With regard to the advantages derived from the hollow and strut—which, 
strange to say, are disputed in some quarters—an experience of several 
years in carriage manufacture and in the conduct of various prolonged and 
