274] EMANANTS AND PITCH INVARIANTS. 289 



274. A Pitch Invariant. 



Let h lt ... h 6 be the direction angles of any ray whatever with regard to 

 six co-reciprocal screws of reference, then the function 



U = ! cos h-i + . . . + a s cos h s 

 is a pitch invariant. 



For, if we augment the pitch of a by x, we have to write for a lf ... a 6 

 the expressions 



x x 



j + 5 COS ! . . . 6 + COS 6 , 



API Ap Q 



and then U becomes 



! COS A! + . . . + COS h K 



x /cos a a cos ^ cos a cos A, 



~r o I T H -- 



Pi 



but from what we have just proved, the coefficient of x is zero, and hence 

 we see that 



! cos ^ -|- . . . + 6 cos h 6 



remains unchanged by any alteration in the pitch of a. 



If we take three mutually rectangular screws, a, /3, 7, then we have the 

 three pitch invariants 



L = 1 cos ^ + . . . + 6 cos a 6 , 

 M = #! cos &J + ... + # 6 cos 6 6 , 



N = O l COS Ci + . . . + # 6 COS C 6 . 



It is obvious that any linear function of L, M, N, such as 



fL + gM+hN, 

 is a pitch invariant. 



We can further show that this is the most general type of linear pitch 

 invariant. 



For the conditions under which the general linear function 



^0,+ ... +A n n 

 shall be a pitch invariant are that equations of the type 



A l cos a, A 6 cos a a _ e 



-- r . . . + - = U ; dec. 



Pi P 



shall be satisfied for all possible rays. 



Though these equations are infinite in number, yet they are only equi 

 valent to three independent equations ; in other words, if these equations 

 are satisfied for three rays, a, b, c, which, for convenience, we may take to 

 be rectangular, then they are satisfied for every ray. 



B. 19 



