504 -20- 



Thence, applying tne criterion of paragraph A.ii to Figure 5, we have: 



The slope of the 3 curve is always^ Y sc that the motion is always of Type M, anfl in 

 particular: 



. s"„ = \- \ ^ S (»«) 



(b) w^ > ^ 



The motion will be of Type II until th.. point A whore S = Y and will then change to Type I, 

 the S curve then boccminj the linv aC of slcpe Y in Figure 5; AC is therefore the tangent at A 

 to tht S curve- since S is continuous.^ The motion will revert to Type M at the point C and^will 

 remain of this type since tne fljpe 5 is d'.creasing steadily and is < Y on CD for all passible 

 positions -f C, 



If the point A corresponds to: 



= a t = e^ (97) 



then from (92) ^^ 'S iliven by: 



la^ S^ (Cos 5^ - Cos 2 6^) = Y (9B) 



and similarly if the point C corresponds to: 



e = a t =9^ (99) 



then It is easily shewn from the tangency property of AC and the equation .(9l) for 5 that: 



Sin0^ (1 - Cos(9^) - Sin(9^ (1 - C'.s^,) = (5^ - (9^) (C:^£ 6^^ - Cos 2^ J (lOO) 



Thence from (70), (90) and (91) it follows th^t: 



!jP = 1 - ^ (1- Cos 5^2+ 5^ (1 - Cos 6?^) 2 

 ^m 



(101) 

 * I {0j^- 0^) X { Sin e^ (1 - Cos 0^) + Sin fl^ (l - Cos ^) } 



for W„ > *,. 

 m z 



The first two terms in (lOi) correspond to the area under CO, the third term to the area 

 under OA and the remaining terms to the ara.i under AC, 



Further, from (9J), (95) anO (98), it follows thjt: 



(102) 



!i] ■= I (Cos 5^ -Cos 2^^) 

 ■ m J 



f;r W > w the corresponding r^nge of 5, being: 



Cos"^ ^ ^ 0. > (103) 



From (lOO), (lOl), (l02) and (96) we can thus evaluate S' /S^ as a function of \/^2' '" 

 the calculations for W > w it is mcst convenient to regard as the basic parameter and a. as an 

 interned i^ite parameter given by (lOC). The only involved part of the calculations Is this 

 calculation of 5, since (lOO) is an implicit equation for 5 ; this evaluation was performed by 

 obtaining a first approximation graphically and then correcting this by (lOO). 



Thus we plotted the curve of oraimte y (say) and abscissa S, 



Y = Sin 5 (l - Ccs i9) (104) 



corresponding 



