350 Mr. G. H. Bryan on the Equations of 



It will be readily found that the six components of the 

 screw motion are 



2j=Vcos# + nasin0 ; jt?s=flcos0, "] 

 t>=0, q = 0, ; . . (37) 



w=Vsin0— Oacos0, r=Osin0, J 

 so that the equations (34) now give 

 X=0, L = 0, I 



Y=nHsin0, M=VHsin0-QaS cos + X2A sin 0, [.(38) 

 Z = 0, N = 0, J 



The impressed wrench therefore has for its axis the shortest 

 distance between the axis of the screw motion of the solid 

 and the axis of the impulse of the cyclic fluid motion. To 

 find the pitch of the wrench, we have, by division, 



- = --acot0+ E , 



that is, 



P = w-acot0 + II (39) 



15. In the case of a fine massless circular ring A vanishes, 

 or the impulse of the cyclic motion is purely translational. 

 For it is clear that the axis of the ring is the axis of this 

 impulse (the above axis of as), also the fluid motion will 

 evidently be unaffected by rotating the ring about its axis; 

 and therefore the modified function is independent of the 

 angular velocity p. 



The equations (34) now become 



X=0, L = 0, -\ 



Y=tB, M=wS, [ ... (40) 



Hence a constant force Y along the axis of y causes 

 uniform rotation with angular velocity Y/E about the axis 

 of z 9 and a constant couple M about the axis of y causes 

 uniform translational velocity M/S along the axis of z. 



It is to be noticed that the impressed wrench never does 

 work in the resulting screw motion, in accordance with the 

 principle of Conservation of Energy. 



16. The above results show the effective forces produced by 

 circulation of the fluid on any perforated solid whatever. In 

 the general case the modified function contains the quadratic 

 terms 37 + X± in addition to the terms of the first degree con- 

 sidered in the above investigation. If we suppose that the 

 solid is moving in any given manner, the six equations of 

 motion (19, 20) determine the components of the impressed 

 wrench (X, Y, Z, L, M, N) necessary to maintain the given 



