XiAh}.— Con tn'/iui./on.'i to Ihc Thcurij nj hicrctvs. 33 



in which 



2m,, = (/-, + p,) COS (03 - 0.) - d2, sin (03 - 0.), ' 



2^31 = {ih + Ih) COS (03 - 0,) - d,, sin (03 - 0,), - (26) 



2ra,= = {ih + ZJs) cos (0. - 0,) - d,, sin (0, - 0,). ^ 



The three equations (25) are obtained immediately by expressing that 

 three neutralizing twists on 1, 2, 3 respectively can do no work against a 

 wrench on 1, or on 2, or on 3. 



In the first group of equations (25) the quantities 



sin (03 - 00 sin (0, - 03), sin (0= - 0,), 



being the amplitudes of the three twists which neutralize, are formed by 

 taking the angles cyclically. For this purpose, the planes in which the 

 different vector-screws lie are not material. 



It might hastily be assumed that the quantities 



sin (03 - 0.), sin (03 - 0,), sin (0, - 0,), 



which occur in the equations (26) as the coefiicients of d,^, d,,, d,2, should 

 also be formed by taking the angles 0,, 02, 03 cyclically. But to do so would 

 have made the formula erroneous. The angles here involved are in each 

 case the right-handed angles between the corresponding pair of vector-screws. 

 The order of superposition of these vectors — that is to say, the relative posi- 

 tions of the planes in which they lie — have to be carefully attended to. It 

 will, of coui'se, be remembered that in obtaining the expression of the virtual 

 coefficient, it was particularly specified that the angle introduced into the 

 expression must invariably be the right-handed angle between the two vector- 

 screws. 



Bight-handed and left-handed Pairs of Lines. 



A pair of lines which do not lie in the same plane and are not at right 

 angles may be distinguished as right-handed or left-handed. 



A disc LM, being supposed to be inserted between the two lines AB and 

 A'B' in fig. 7, and between PQ and P'Q' in fig. 8, enables us to represent that 

 AB lies over A'B', and that PQ lies over P'Q'. 



As a convenient mnemonic we may fancy the right arm AB crossed over 

 the left A'K to form fig. 7, and the left arm QP crossed over the right Q'P" 

 to form fig. 8. Thus we may appropriately distinguish the two figures as 

 right-handed and left-handed, so long as the two lines of each paii" do not 

 intersect, and so long as the angles AOA' or POF are both acute. Suppose 

 L AOA' was increased up to 90°, then a critical stage is reached ; and the 

 distinction between a right-handed pair- and a left-handed pair will at that 



[5*] 



