49] SCREW CO-ORDINATES. 43 



Hence by subtracting the several formulae (i) from the formulae (ii) we 

 obtain 



// ( 5 + ) - z (flf g + 4 ) = a (a, - a,) - a (0, - 0,)j 



z n (! -I- a,) - a? (a 5 + a fi ) = 6 (a, - a 4 ) - b (0 3 - 4 ) L ........ (iiii). 



# ( 3 + 4&amp;gt; - 2/o (i + 2 ) = c ( s -- a 6 ) - c (0 5 - 6 ) ) 

 The six e( {nations (iii) and (iiii) determine 6 l) ... # in terms of ,,.... 



49. Principal Screws on a Cylindroid. 



If two screws are given we determine as follows the pitches of the two 

 principal screws on the cylindroid which the two given screws define. 



Let a and /3 be the two given screws. Then the co-ordinates of these 

 screws referred to six canonical co-reciprocals are 



!,... and fr,...fr. 



The co-ordinates of any other screw on the same cylindroid are propor 

 tional to 



pa, + fr, py 2 + /3,, . . . pa, + & ; 



when p is a variable parameter. 



The pitch p of the screw so indicated is given by the equation ( 41) 

 a (pa, + J3J* - a (pa, + /3,) 2 + b (pa 3 + &)* - b (pa, + &) 3 



= p [{p ( ttl + a,) + fr + fr}* + p {(a, + a 4 ) + & + /3 4 j 2 + p {(a s + a) + fr + ft,]- 2 ], 



or 



p*p a + 2pvr ali + p ft = p {p 2 + 2p cos (a/3) +1], 

 or 



p- (p a -p) + 2p {CT a/3 - p cos (a/9)] + pp p = 0. 



For the principal screws p is to be determined so that p shall be a maxi 

 mum or a minimum ( 18), whence the equation for p is 



Ka0 - p cos (a/3)] 2 = (p a - p) (p ft - p), 

 or 



I? sin 2 (a/3) + p (2OT a/3 cos (a/8) - p a - pp) + p a p ft - vr^ = 0. 



The roots of this quadratic are the required values of p. 



The quadratic may also receive the form 



= (P ~ Pa) (p ~ Pft) sin 2 () + d a/5 -sin (a/3) cos (a/9) (/j a 



- i (Pa -p p y cos 2 (a/S) - i #. sin 2 (a/3), 

 where d a p is the shortest distance of a and /3. 



