270] EMANANTS AND PITCH INVARIANTS. 285 



269. Property of the Pitches of n Co-reciprocals. 



The theorem just proved can be extended to show that the sum of the 

 reciprocals of the pitches of n co-reciprocal screws, selected from a screw system 

 of the nth order, is a constant for each screw system. 



Let A be the given screw system, and B the reciprocal screw system, 

 Take 6 n co-reciprocal screws on B, and any n co-reciprocal screws on A. 

 The sum of the reciprocals of the pitches of these six screws must be always 

 zero ; but the screws on B may be constant, while those on A are changed, 

 whence the sum of the reciprocals of the pitches of the n co-reciprocal screws 

 on A must be constant. 



Thus, as we have already seen from geometrical considerations, that the 

 sum of the reciprocals of the pitches of co-reciprocals is constant for the 

 screw system of the second and third order ( 40, 176), so now we see that 

 the same must be likewise true for the fourth, fifth, and sixth orders. 



The actual value of this constant for any given screw system is evidently 

 a characteristic feature of that screw system. 



270. Theorem as to Signs. 



If in one set of co-reciprocal screws of an n-system there be k screws with 

 negative pitch and n k screws with positive pitch, then in every set of 

 co-reciprocal screws of the same system there will also be k screws with negative 

 pitch and n k screws with positive pitch. 



To prove this we may take the case of a five-system, and suppose that of 

 five co-reciprocals A l , A 2 , A 3 , A 4 , A 5 the pitches of three are positive, say 

 mf, ra 2 2 , TO/, while the pitches of the two others are negative, say ra 4 2 , 



Let 8 be any screw of the system, then if 1} ... # 5 be its co-ordinates 

 with respect to the five co-reciprocals just considered, we have for the pitch 

 of 6 the expression 



raM 2 + w 2 2 2 2 + ra 3 2 &amp;lt;9 3 2 - m 4 2 4 2 - m 5 2 5 2 . 



Let us now take another set of five co-reciprocals B 1} B 2 , B 3 , B 4 , B 5 

 belonging to the same system, then the pitches of three of these screws must 

 be + and the pitches of two must be . For suppose this was not so, but 

 that the five pitches were, let us say n^, n^, w 3 2 , v? 4 2 , w s 2 . Let the co-ordinates 

 of with respect to these new screws of reference be &amp;lt;f&amp;gt; 1} &amp;lt;f&amp;gt; 2 , ... B , then 

 the pitch will be 



V& 2 + n.*&amp;lt;f&amp;gt; 2 2 + n s a 3 2 + nty* - n 5 2 &amp;lt;/&amp;gt; 5 2 . 



