Ball — On Properties of the Cylindroid. 519 



convenience of demonstration we have regarded the forces as consez'- 

 vative ; but it is to be I'emembered that the condition of reciprocity is 

 a purely geometrical one, involving only a certain metrical relation 

 between the positions of the two screws and their pitches. "We adduce 

 the case of a conservative system of forces merely to show that this 

 condition must be observed between 6 and y. This being so, the re- 

 lation of reciprocity is true whatever be the forces which constitute 

 the wrench. 



"We proceed to study the consequences of the reciprocity of 6 to 

 the group of screws typified by y, lying on the surface 8. Let there be 

 four screws, 6i, 0o, O3, Oi, drawn reciprocal to any two screws of S : 

 since a screw is defined by five conditions, it is plain that a screw which 

 fulfils the four conditions of being reciprocal to Oi, $2, 0^, 0^ will have one 

 degree of freedom, and must, therefoi-e, be confined to a certain ruled 

 surface. This surface must, of course, include S, as all on the screws 

 on it are reciprocal to 61, O2, O., Oi; further, it cannot include any screw 

 not on S ; for suppose it did contain a screw e, then as e and ani/ screw 

 y on S are reciprocal to ^1, O2, O3, Oi, it will follow that any screw on 

 the surface made from e and y, just as S is made from a and f3, must 

 also be reciprocal to ^1, 0^, O3, Oi. As y may be selected arbitrarily on S, 

 we would thus have the screws reciprocal to Oi, O2, O3, Oi limited not 

 to one surface, but to a whole group of surfaces, which is impossible. 

 It is therefore the same thing to say that a screw lies on S, as to say 

 that it is reciprocal to Oi, O2, O3, Oi. 



Since the condition of reciprocity involves the pitches of the two 

 screws in an expression containing only their sum, it follows that if 

 all the pitches on Oi, Oo, O3, 6^ be increased by - m, and all those on 8 

 be increased by + m, the reciprocity will be undisturbed. Hence, if 

 the pitches of all the screws on 8 be increased by + m, the surface so 

 modified will still possess the property, that twists about any three 

 screws will neutralize each other if the amplitudes be properly 

 chosen. 



"We can now take a step in our study of 8, and show that 

 there cannot be more than two screws of equal pitch thereon ; for 

 suppose that there were three screws of pitch m, if we then apply 

 the constant -w to all, we shall have on 8 tkree screws of zero pitch. 

 It must therefore follow that three forces on 8 can be made to neu- 

 tralize ; but this is obviously impossible, unless these forces intersect 

 in a point and lie on a plane. In this case the whole of 8 degrades to 

 a plane, and the case is a special one devoid of interest for our present 

 purpose. It will, however, be seen that in general 8 does possess 

 two screws of any given pitch, for it is well known that a wrench 

 can always be decomposed into two forces in such a way that the 

 line of action of one of these forces is arbitrary. Suppose that 8 

 only possessed one screw X of pitch m. Eeduce this pitch to zero ; 

 then any other wrench must be capable of decomposition into a force 

 on X ('/. e. a wrench of pitch zero), and a force on some other line which 

 must lie on 8 ; therefore in its transformed character there must be a 



