BODY BY THE AID OE THE THEORY OE SCREWS. 
19 
To express this equation concisely we introduce two classes of subsidiary magnitudes. 
We write one magnitude of each class as a determinant. 
7i, iffi+xA yi a \-> 
«i, 
01, 
7l 
£ 2 0 2 +z 2 « 2 — a- 2 y 2 , f 2 y 2 +# 2 0 2 — y 2 a 2 , 
«2, 
02, 
72 
§ 303 + ^3^3 X 3 y 3 , ^ 3^3 “l - ^303 ^ 3^35 
« 3 , 
03, 
7s 
eA + « 4«4 — ^474 + ^404—^, 
« 4 , 
04, 
74 
g 5 fi 5 +z 5 a 5 — Xtfs, g s y s +x& s —y 6 et ? . 
«5, 
05, 
75 
By cyclical interchange the two analogous functions Q and B, are defined. 
§101+*1«1— ®i7i, 
§i7i+#i0i“ 
?1«1, 
01, 
7i 
§202 - l - ^2 a 2 *^272, 
§272 + ^202 3^2 a 2, 
§2«2, 
02, 
72 
§ 303“i“^3 a 3 *^373? 
§373+^303 ^3 a 3, 
§3 a 3, 
03, 
73 
§404 + ^4 — ^ 74 , 
§474 + ^404—2/4^4, 
§ 4 « 4 , 
04, 
74 
g 5 0 5 +Z 5 a 5 — # 5 y 5 , 
§575+^505 — «/ 5 « 5 , 
§ 5 « 5 , 
05, 
75 
By cyclical interchange the two analogous functions M and N are defined. 
The equation for g reduces to 
(P 2 +Q 2 +R 2 )e+PL+QM+RN=0. 
The reduction of this equation to the first degree is an independent proof of the 
important principle, that one screw and only one can be determined which is reciprocal 
to five given screws ; g being known, a, 0, y can be found, and also two linear equations 
between x J , y', z', whence the reciprocal screw is completely determined. 
7. Coreciprocal screws. — A set of six screws can be chosen, so that each screw is 
reciprocal to the remaining five. For take A 2 arbitrary; A 2 reciprocal to A 2 ; A 3 reci- 
procal to A 1? A 2 ; A 4 reciprocal to A„ A 2 , A 3 ; A s reciprocal to A„ A 2 , A 3 , A 4 ; and A 6 
reciprocal to A„ A a , A 3 , A 4 , A s . A group constructed in this way is called a set of 
coreciprocal screws. 
Thirty constants determine a group of six screws. If the group be coreciprocal, fifteen 
conditions must be fulfilled ; we have therefore fifteen elements still disposable, so that 
we are always enabled to select a coreciprocal group with special appropriateness to the 
problem under consideration. 
The facilities presented by rectangular axes for questions connected with the dynamics 
of a particle have perhaps their analogues in the conveniences which arise from refer- 
ring the twist coordinates of a rigid body to a group of coreciprocal screws. 
8. Resolution of a wrench along six coreciprocal screws. — The resolution of a wrench 
S (or of a twist velocity) into six wrenches (or six twist velocities) of magnitudes X 2 &c., 
X 6 along six reciprocal screws of reference A 2 &c., A 6 is thus effected geometrically. 
Draw the cylindroid ( A 15 A 2 ), select on this cylindroid the screw P reciprocal to S [art. 44] ; 
if a rigid body only free to twist about P be acted upon by wrenches about S, A 1} &c., 
D 2 
