266] EMANANTS AND PITCH INVARIANTS. 279 



This can be verified in the following manner. We have 



V = p\&quot;\ 

 dv / d 



and, therefore, if X be reciprocal to the first screw of reference, the formula 

 to be proved is 



A few words will be necessary on the geometrical signification of the 

 differentiation involved. Suppose a Dyname A, be referred to six co-ordinate 

 screws of absolute generality, and let us suppose that one of these co 

 ordinates, for instance X 1; be permitted to vary, the corresponding situation 

 of X also changes, and considering each one of the co-ordinates in succession, 

 we thus have six routes established along which X will travel in correspond 

 ence with the growth of the appropriate co-ordinate. Each route is, of 

 course, a ruled surface ; but the conception of a surface is not alone adequate 

 to express the route. We must also associate a linear magnitude with each 

 generator of the surface, which is to denote the pitch of the corresponding 

 screw. Taking X and another screw on one of the routes, we can draw a 

 cylindroid through these two screws. It will now be proved that this 

 cylindroid is itself the locus in which X moves, when the co-ordinate cor 

 related thereto changes its value. Let 9 be the screAV arising from an 

 increase in the co-ordinate Xjj a wrench on 6 of intensity 6&quot; has components 

 of intensities #/ , . . . 6 &quot;. A wrench on X has components X/ , . . . X 6 &quot;. But 

 from the nature of the case, 



If therefore & be suitably chosen, we can make each of these ratios 1, 

 so that when 6&quot; and X&quot; are each resolved along the six screws of reference, 

 all the components except $/ , X/ shall neutralize. But this can only be 

 possible if the first reference screw lie on the cylindroid containing and X. 

 Hence we deduce the result that each of the six cylindroids must pass 

 through the corresponding screw of reference ; and thus we have a complete 

 view of the route travelled by a screw in correspondence with the variation 

 of one of its co-ordinates. 



Let the six screws of reference be 1, 2, 3, 4, 5, 6. Form the cylindroid 

 (X, 1), and find that one screw 77 on this cylindroid which has with 2, 3, 4, 5, G, 

 a common reciprocal ( 26). From a point draw a pencil of four rays parallel 

 to four screws on the cylindroid. Let OA be parallel to one of the principal 

 screws ; OX be parallel to X, Otj to 77, and Oh to the first screw of reference. 



