THE SLIP OP THE DRIVING-WHEEL OF A LOCOMOTIVE ENGINE. 
565 
and 
W/ifff (9+ cos fl'l TTrf. I 
Y= — r- cos 0 +^^ — TT =W 1+---— , 
a 
whence we obtain 
Y,=W,-fW H 
and 
h(xi^ cos fl 
9 
X_ « \ 
Yi 
a 1 g{k^-\-a^) ( aco^l q{k‘^ + a^) | 
g{k^ + a^) 
Wi + W 1 + 
Assuine 
ho)^ cos I 
9 
(' 
^ W 
(63.) 
(64.) 
+ COS ( 
sin 0 
cos 3 
. / W, \ q , sir 
l + / 3 cos 5 d’^u {— / 3 (/ 3 + cosfl) + 2 ( 1 +/3 cos 6 )} sin 0 
du 
d^ (j3 + cos 
(/3+ cosfl)' 
Now if (3> ], there will be some value of & for which cos^=0, and therefore 
^2 
du ^^u 
l-}-/3cos^=0 ; and since for this value of 6, ^=0, and 
c?9““ ’ (/32 — i)t 
X 
it follows that 
it corresponds to a maximum value of u, and therefore of y* 
But if (3 < 1, then there is some value of cos d for which jS-f- cos ^=0, and therefore 
for which M=infinity, which value corresponds therefore in this case to the maximum 
or y' 
Thus then it appears that according as 
(3»‘-(l+w)£-«>‘«‘-“’<f(‘+^) (65.) 
X 
the maximum value of y- is attained when cos^=— /3 or = — that is, when 
a / W \ 
cosfc-^(l+^j »■■ = 
ha? 
( 66 .) 
X 
In the one case the maximum value of ^ will be infinity, (6/.) 
and in the other case it will be represented by the formula 
Y{ 
1 
k^[g 
g{k^ + a^) / 
( 68 .) 
In the first case, i. e. when |3< 1, the w.heel will slip every time that it revolves, what- 
ever may be the value of f. In the second case, or when |3 > 1, it will slip if f do not 
exceed the number represented by formula (68.). The conditions (65.) are obviously 
MDCCCLI. 4 D 
