Ball — Contributions to the Tlieorij of Screws. 31 



Thus we leani that when tliree twists on tlirce vector-screws neutralize, 

 the amplitude of each twist is proportional to the sine of the angle between 

 the other two.* When the signs of the amplitudes are also required, tlie 

 formula (14), taken in conjunction with fig. 6, must also be attended to. 



It remains to investigate the way in which the three vector-screws 

 are related. It will be convenient for this purpose to suppose that the two 

 screws a and /3 are given so that i\, p^, 0^, 0^, 2„ z^ are all known ; and 

 we shall seek the equations connecting these quantities with jiy, 0y, z^. As 

 7 is now regarded as a current vector-screw, we shall write ji?, 6, z for 2\, 0y, ~r 

 We can eliminate z by multiplying (10) by cos d and adding it to (11), 

 after multiplication by sin 6. If at the same time we substitute for a, /3', 7' 

 from (1-4), we obtain 



p sin (Op - 6'„) = -H Pa cos (6'„ - 6) sin (6^ - 0) 

 + Pp cos {9^-6) sin {0 - 0,) 

 + s. sin {0, - 0) sin {0^ - 0) ^ ' 



+ zp sin (0p - 0) sin {0 - 0„). 



If we introduce three new quantities. A, B, C, which are constant so 

 far as 2^ ^^^ ^ ^i'® concerned, and defined by the formulae 



A^- Pa sin 0^ cos 0p + pp cos 0„ sin 0^ + {z^ - z^) cos 0^ cos ^,3, (16) 



2B ^ (pp - Pa) cos {0a + 0^) + {z^ - Za) sin (^„ + 0^), (17) 



C = + Pa cos 0a sin 0p - pp cos 0^ sin 0a + (j„ - z^) sin 0a cos 0^, (18) 

 with this substitution we may write equation (15) as follows 



p sin {0^ -0a)== A sin= + 2B sin cos 6* + C cos^ 0. (19) 

 If ^ be a maximum or minimum, then of course 



{A - C) sin 20 + 2B cos 261 = 0. (20) 



There are thus two values of differing by 90°, of which each gives a 

 stationary value of 2). We shall take these screws of stationary pitch for 

 a and /9, and we shall so adjust the line from which is measured that 

 ^„ = and 0^ = 90°. The formula (20) must thus reduce to sin 20 = 0, 

 whence £ = 0. If we substitute in the general expression for 2B, (17) we 

 obtain 



«3 - 2« = 0, (21) 



from which we learn that the two screws of stationary pitch in the cylin- 

 droid, or in what we may also call a two-system of screws, intersect at 

 right angles. 



The general expressions also give in tliis ease A = pp and C = p^, 



* "Treatise," p. 21. 



R.I. A. PROC, VOL. XXVIU., SECT. A. [6] 



