A Treatise on the Theory of Screws

470 THE THEORY OF SCREWS. [425 425. The Vector in Orthogonal Co-ordinates. Since, in general, a u cos t/ = — , 1 z we have for the vector (§ 419) the following conditions:— (11) = (22) = (33) = (44), and also, (12) + (21) = O, and the similar equations. In fact, U can only differ from fl by a constant factor. The orthogonal equations require the following conditions— + (11). (12) - (12) .(11) + (13) . (23) + (14) . (24) = 0, + (11). (13) - (12). (23) - (13). (11) + (14). (34) = 0, + (11). (14) - (12). (24) - (13) . (34) - (14) . (11) = 0, + (12) • (13) + (11). (23) - (23) . (11) + (24). (34) = 0, + (12). (14) + (11) . (24) - (23). (34) - (24). (11) = 0, + (13) • (14) + (23) . (24) + (11). (34) - (11) . (34) = 0, + (ll)2 + (12)“ + (13/ + (14)2 = 1, + (12)2 + (11/ + (23)2 + (24)2 = 1, + (13)2 + (23)3 + (11)“ + (34)2 = 1, + (14)2 + (24)2 + (34)2 + (11)2 = ! We now introduce the notation:—■ (11) = a; (12) = /3; (13)= 7; (14) = 8, and the equations give us + 7(23)+ 5 (24) = 0................................................(i), Æ (23) + <5 (34) = 0 ................ (ii), -£(24)- y (34) = 0 ....................(iii), + ßy + (24)(34) = 0.....................(iv), + /3S - (23) (34) = 0 .............. (v), + 78 4- (23) (24) = 0 ............. (vi), + a2 + /33 + + 52 = j + /92 + a2 + (23)2 + (24)2 = 1 + y2 + (23)2 + a? + (34)2 =1..................... + 62 + (24)2 + (34)2+ a2 =1,