f: ilsd 



ON THE STRUCTURE OF COVERED WAY3. "1545 



sine of the angle DB C being ?-£ and its cosine |^, and 



tlie sine and cosine of A B C "being —, and ^, the ^y^e 



of DBA is E2_i£i^ + ££A£,'and cpnsequ^nily^ ETlfe 



=: BD . D<^-A.B-t-gD A;C ^ ^^1^;^,^^ divided by A C, is 

 B C q 



r> n T^ C . A B 5-2Ji, and this must be a maximnm, con- 



AC.BCq^ B U q 



sequently, B C bemj^ supposed constant, —- . • ■■ 4-- ,, ;^ 



must also be a maximum. Then if we make D F perpen- 

 dicular to B C, and the angle F D G =i A C B, D F will 



be 5^, F G = :^. ^££, and B F ^ i^, so that B G "X 



B C AC BC ^>^ ' 



must be a maximum, which will evidently happen wh^ti 

 DG is a tangent to the semicircle BD G, and the angle 

 D B C half D II C, which is the difference between ABC 

 and a right a-rigle. If we wish to determine tlie ^oj>ortit)n Case of the 



of the friction to the pressure when tlie friction is barely ^J^^^f^ ^^^^ 



*■ ..... bureiv sum- 



capable of retaining the block in its situation in the most ci nt to retain 



unfavourable position, let x be the sine, and y the cosine 6f ^'^'^ ^^ock. 

 hAlf the angle A B C, then the sine and cosine of half a 

 right angle being -v/ J-, the sine of A B D or BD E, as 

 well as that of B CD, will be a/ | ;r + 1/ f «/. Now, if 

 the weight be B D, B C zz !iiL_, and the sine of ABC 



being 2xt/, A C is "^^^ but the weight which pro- 



duces the friction is three times the weight of a single block, 

 the friction on the upper surface being derived from the 

 pressure of the highest block, and that on the l«wer frOm 

 the pressure of both blocks; while the tendency to descend 

 belongs to the lower block only, and is therefore expressed 

 by B D. -y/ 1^ [x + y) ; hence we have the equation V, | 



(.1- + 1/) - -7 r/-^V-^ ' tlierefore i {x + i,'] ^ - G x y, 

 V 2 V'^ -r y) 



[x 4- y)2 — I'i xy, x^ 4- y^ = lO'J^y ==: Ij 2a:y — i, which 



is the sine of A B C, and the friction is in this case to .the 



oblique resistance as 1 to 5, and to the pressure nearly as The frictioh 



10 to 47 : so that whenever the friction is ereater than this, sufficient to re- 



II taui two pairs 

 which 



