60 Pfoceedinffs of tJie Royal Irish Acaihmj/. 



The desired relations are proved by making use of the well-known 

 property that the screw reciprocal to five screws, a, |3, 7, S, t, will retain 

 the same relation to a, (3, y, S, t if, instead of pe, the pitch of 6, we write 

 Pe - k ; while, instead of jj., 2^^, Py, Ps, Pt, we write ;;„ f k, p^ + k, p^ + k, p^ + k, p, + k 

 respectively, where k is any scalar ; this follows at once from the fact that, if 

 6 is reciprocal to a, 



(Pe + I'a) cos 9^ - d sin 0, = 0, 



where tf„ is the right-handed angle between B and a ; but of course this 

 equation may equally be written thus 



l(i'» - ^0 + O^a + k)\ cos Oa -d sin 0. = 0. 

 If tlie pitch of the screw with vector coordinates {n, A) be increased by k, but 

 without any other change, then the coordinates merely become |(/u + AX), Xj. 

 This is obvious from the fact that 



FCm + AX)/X = V^IX and S (ji + kX)/X = S^/X + k. 

 If the pitches of the five screws be increased by k, but no other change is 

 made, then their vector coordinates become 



I Oil + A-X,), X.l ; . . . K/u, + kXi), X.|. 

 If we substitute {/i, + AX,) . . . Oi, + kXi) for /i, . . . juj in the expression for 

 X in (91), and denote by X, the value which X then assumes, we have, as is easily 

 seen, X. = X + A-^ + k'B, (98) 



where A and B are the vectors in (95) and (96). 



In like manner when the same substitutions are made in (92) we have for 

 fik the value which /u then assumes 



,xk ' ix - Dk ^ Cn<? - Bk*. (99) 



But, as just pointed out, the screw (jn,, X*) can only differ from the screw 

 {p, X) in that the pitch of the second is A- less than the pitch of the first. 

 It follows that liiJi, + kXjt), X*! is a screw identical with (/u, X). Thus (fi, X) 

 and I l/x + A: (X - D) + A:* (-4 + C)|, (X + kA + k'B)] must be the vector 

 coordinates of one and the same screw whatever be the value of k. Hence 



we must have 



A = 0, B = 0, C = 0, D = X. 



The^e properties of the vector expressions are of course easily verified 

 by direct calculation. 



We see that -4 = by writing separately the terms involving /u„ which are 



Vn: iXiSXJ^X - X.SXtXX + X,SX^»X, - XsSXiXjX.). 

 But from te known quaternion formula the quantity in the bracket is zero. 

 In like manner each of the other groups of terms is zero. Thus A is verified, 

 and this includes .B = by interchanging /i and X. 



