40 Proceedings of the Royal Irish Academy. 



7\ 7^ f) 



III. — On the Pitch Operator —- + ;^ + • ■ • + ;:r- • 



Let 01, 02, ■ ■ • 0« be 7i-screws belonging to an (n - l)-sj-stem. If 

 0i' 0/ . . . 0,,' be the corresponding amplitudes of ?! -twists which neutralize, 

 then the work done by a wrench on any screw r] in the course of the 

 appUcation of these ?i -twists must be zero, and consequently 



0.'^^, + 0:-:=^ + + ^.'^ft., = 0. (28) 



Indeed the necessaiy and sufficient condition that the ?!-screws shall belong 

 to an (71 - 1) -system may be expressed by saying that in such a case it 

 must be possible to find a system of quantities, 0/, 0i, . . . 0„', independent 

 of -q, which shall make this equation true for all possible screws ij. 



If (28) is to be satisfied for all screws »;, it must of course be satisfied ; 

 if while 1) remains otherwise unchanged, we change jy^ into p^ + h, where 

 A is any linear quantity. 



As p^ only enters into the virtual coefficients w#„ in the several com- 

 binations {p, + p^), (p,+ p,), . . . (/), + p,), it is plain that the efiect of 

 changing ;>, into }\ + h is just the same as to leave />, unaltered, but 

 to change p,, pt . . . Pu into (j>, + h), (jh + h), . . . {p„ + h] respectively. 



Thus we have a result already well known in the theory of screws,* 

 that if ^1 ... ^, be n-screws belonging to an (n - l)-8ystem, they will 

 still belong to an [n - 1)- system if these pitches pi, . . . p„ be each increased 

 by the same quantity h, where h may have any value whatever. 



As the simplest example we recall that if jj be the pitch of a screw on a 



cylindroid, we know that 



p = }h cos' B + pt sin' 0, 



which may of course be written thus 



/> + A = (/), + /i) cos' + {p, + h) sin* 0. 



We can now define the pitch-operator 



and from what we have just proved, it follows that if /* = be any general 

 equation connecting n-screws, which belong to an {n - 1) -system, then 



^F = 0. 

 If this operation A is applied to w„, the virtual coefficient of two screws 

 (1) and (2), it gives 



Ats„ = cos (121, 



» " Treatise," p. 2S8. 



