90 
Proceedings of Roy cd Society of Edinburgh, [jan. 31 , 
differentiated with respect to 0 and 
da 
db 
put equal to zero in the result 
— i.e. (5) must he differentiated partially with respect to 0 . There 
then results the condition 
_ s sin /3 
an a s cos {3 sin 1 2 0 — cos 2 0 
• ( 6 ) 
In strictness, this step ought to he immediately justified by an 
examination of the sign of , under the condition (6). This will 
he found to involve a considerable amount of labour ; and, since the 
operation is essentially of the nature of a verification, the author 
proposes to accept, for the present, (6) as the condition for a a 
maximum, and afterwards to justify whatever assumption this may 
involve, by a process of verification. The mere fact, however, of 
obtaining a result which does not necessarily imply that a must be 
zero, indicates the existence of a maximum or minimum condition 
different from a = 0 ; which is, of course, the condition which a 
fulfils when it is a minimum as regards mere numerical magni- 
tude. Farther, from the theory of stress, it is known that, corre- 
sponding to the maximum value of a, there is a maximum value 
of equal numerical amount hut of opposite sign, 
How the maximum value of a is to he <f>, the angle of repose ; the 
angle, that is, whose tangent is the coefficient of friction of earth 
upon earth, for earth of the given kind. 
Let 0 1 be the value of 6, when a = cj>. Then 0 = 0 X and a = cf> must 
satisfy (5) and (6) simultaneously. Therefore — 
1 + s sin /? tan # x 1 + tan tan 0 l ^ 
s (cos/5 + sin/? cot 0 X ) 1 — tan </> cot 0i 
and 
tan (j) 
ssin /3 
scos/3 sin 2 — cos 2 #! 
(8) 
(8) may be written 
, 1 + s sin R cot <f) /qx 
tan 2 #! = — t 7 > n 7 -T x ..... (9) 
s (cos fj — sin p cot <p) 
Eliminating 6i between (7) and (8) there is obtained finally, after 
solving for s, 
