282 



THE THEORY OF SCREWS. 



[267, 



and we learn that this expression will remain absolutely unaltered provided 

 that we only change from one set of co-reciprocals to another. In this / is 

 perfectly arbitrary. 



268. Property of the Pitches of Six Co-reciprocals. 



We may here introduce an important property of the pitches of a set of 

 co-reciprocal screws selected from a screw system. 



There is one screw on a cylindroid of which the pitch is a maximum, 

 and another screw of which the pitch is a minimum. These screws are 

 parallel to the principal axes of the pitch conic ( 18). Belonging to a 

 screw system of the third order we have, in like manner, three screws of 

 maximum or minimum pitch, which lie along the three principal axes of 

 the pitch quadric ( 173). The general question, therefore, arises, as to 

 whether it is always possible to select from a screw system of the ?tth order 

 a certain number of screws of maximum or minimum pitch. 



Let 1} ... # 6 be the six co-ordinates of a screw referred to n co-reciprocal 

 screws belonging to the given screw system. Then the function p e , or 



is to be a maximum, while, at the same time, the co-ordinates satisfy the 



condition ( 35) 



20! s + 220 A cos (12) = 1, 



which for brevity we denote as heretofore by 



Applying the ordinary rules for maxima and minima, we deduce the six 



equations 



dR 



9 - = 



From these six equations O l , ... 9 6 can be eliminated, and we obtain the 

 determinantal equation which, by writing x= 1 +po, becomes 



1 %PI, cos (21), cos (31), cos (41), cos (51), cos (61) 



cos (12), l-#p a , cos (32), cos (42), cos (52), cos (62) 



cos (13), cos (23), l-a;p 3) cos (43), cos (53), cos (63) 



cos (14), cos (24), cos (34), lasp t , cos (54), cos (64) 



cos (15), cos (25), cos (35), cos (45), lacp s , cos (65) 



cos (16), cos (26), cos (36), cos (46), cos (56), 1 - ay&amp;gt; 6 



= 



