318 



THE THEORY OF SCREWS. 



[298, 



Let 



! a6 l = X ly 



4 4- 6# 4 = X t , 

 c0 6 = X 6 , 



and the equation becomes 



= (! + 2 ) (X, + X,} + ( 3 + 4 ) (X 3 

 and the two other L equations give 



= (A + &) (X, + X 2 ) + (0, + &) (X 3 



= (71 + 7 



5 + 6 ) (X 5 + X 6 ) ; 



/3 6 ) (X, 



a) + (73 + 74) (-^s + -^4) + (75 4- 7 6 ) (X s + X 6 ). 



If we eliminate X^ + X i} X 3 + X t , X S + X S from these equations, we 

 should have 



= Otj + 85 0(3 + 4 Qfg + Ct 6 



A 4/3, & 4- A & 4 

 7i +7s 7s + 74 7s + 7e 



But this would only be the case if a, /3, 7 were parallel to a plane, which is 

 not generally true. Therefore, we can only satisfy these equations, under 

 ordinary circumstances, by the assumption 



In like manner, the equations of the M type give 



Pa^Or, = 0, 



Substituting, in the first of these, we get 



- p a (017! 0j - a?? 2 ^ 2 4- br} 3 3 - 6774 4 + cv) s s - cy^e) = ; 

 which reduces to 



ac^Zj - a 2 Z 2 + ba 3 X s - 6a 4 Z 4 + ca s X s - ca 6 X e = ; 



but we have already seen that X^ -+ X 2 = 0, &c., whence we obtain 

 Xi (a! + aoz) + X 3 (6a 3 + 6 4 ) + X 5 (c 6 + ca 6 ) = ; 



with the similar equations 



X l (a/9, + a/8 2 ) + X 3 (b/3 3 + bfr) + X 5 (cyS 5 + c&) = 0, 

 ^! (ay l + ay 2 ) + X 3 (by, + by 4 ) + X 5 (c% + cy 6 ) = 0. 



These prove that, unless a, @, y be parallel to a plane, we must have 



