PROFESSOR MOSELEY ON THE THEORY OP MACHINES. 299 



(observing that Cg = Sg), or bringing Sg under the radical sign, 



^ { P^^ S,2 L^ + 2 P, S,2 P, M + Pj^ S,2 a,' } *, 

 or 



^-^ { U/ L2 + 2 Uj Sj P3 M + P,^ &/ a,2 } ', 



so that in this case of a constant direction of the moving pressure, and a constant 

 amount and direction of the working pressure, the modulus becomes, 



and the work lost by friction whilst the space Sg is described by the working point, 

 is represented by the term involving the irrational quantity in this equation. 



12. A machine working about an axis of given dimensions under two pressures, 

 Pj and P2, the direction and amount of one of which Pg are given, it is required to 

 determine that constant direction in which the other pressure Pj must be applied, so 

 that the machine may be worked with the greatest economy of power. 



It has been shown in the last section that the work lost by friction is represented, 

 in the case here supposed, by the formula 



^~{W1^'+2V,S,P,M + P/S,^a,'y (17.) 



The machine is evidently worked then with the greatest economy of power to yield 

 a given amount of work, U2, when this function is a minimum. Substituting for U 

 its value 



fli^ + 2 fli a2 cos /i 2 + «2^ 

 and for M its value 



fli {flg cos tyj^ + tti cos 12^} (section 10.), 

 it becomes 



Now let us suppose that the perpendicular distance a2 from the centre of the axis 

 at which the work is done, and the inclination /g^ of its direction to the vertical, are 

 both given, as also the space Sg through which it is done, so that the work is given 

 in every respect ; let also the perpendicular distance a^ at which the power is applied, 

 be given ; it is required to determine that inclination /, 2 of the power to the work 

 which will under these circumstances give to the above function its minimum value, 

 and which is, therefore, consistent with the most economical working of the machine. 



Collecting all the terms in the function (18.) which contain (on the above suppo- 

 sitions) only constant quantities, and representing their sum 



W « + «2') + 2 P3 S2 a,^ (U2 cos /2^ + P3 S2) 

 by C2, it becomes 



^-^ { 2 «l «2 U2 (U2 cos /1.2 + P3 S2 COS S,^) + C2 j *. 

 MDCCCXLI. 2 R 



