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XXXIII. 
A NOTE OK A DETERMmANT IN THE THEORY OE SCREWS. 
By sir ROBERT S. BALL. 
[Read January 13, 1890.] 
It was only quite lately that I discovered some properties of a 
determinant in the "Theory of Screws," which I would like to 
place on record in the Proceedings of the Academy. = 
Let 6i, ... ^6 he the six co-ordinates of a screw referred to a co- 
reciprocal system of screws of reference. 
Then, denoting as the pitch we have 
where 
E = ei' + ... + 0s^ + 20, $2 COS (12) + 20, 0^ cos (13) + &c., 
in which '12) is the angle between the 1st and 2nd screws of reference, 
and similarly for the others. 
Let us now determine the condition that the pitch be a 
maximum. Then, as in ('' Theory of Screws," p. 148), we have, of 
course, by the usual process, 
2p,e,-pe^^ = 0. 
^ n ^ 
2p^0^- Pb-^ = 
Erom these six equations ^i, . . . 0^ can be eliminated, and we obtain 
the well-known harmonic determinant which, by writing x = \ Pq^ 
becomes 
0 = 
1 - Xp,y 
cos (21), 
cos (31), 
cos (41), 
cos (51), 
cos (61) 
cos (12), 
1 - xp2, 
cos (32), 
cos (42), 
cos (52), 
cos (62) 
cos (13), 
cos (23), 
1 - xp^, 
cos (43), 
cos (53), 
cos (63) 
cos (14), 
cos (24), 
cos (34), 
1 - xp^, 
cos (54), 
cos (64) 
cos (15), 
cos (25), 
cos (35), 
cos (45), 
1 - xp^, 
cos (65) 
cos (16), 
cos (26), 
cos (36), 
cos (46), 
cos (56), 
1 -xp^ 
