267] EMANANTS AND PITCH INVARIANTS. 281 



reduce to an identity ; but retaining / in its general form, we can deduce 

 some results of very considerable interest. The discussion which now follows 

 was suggested by the reasoning employed by Professor W. S. Burnside* in 

 the theory of orthogonal transformations. 



Let us suppose that we transform the function f from one set of co- 

 reciprocal screws of reference to another system. Let p 1} ... p 6 be the 

 pitches of the first set, and q lt ... q 6 be those of the second set. Then we 

 must have 



for each merely denotes the pitch of the Dyname multiplied into the square 

 of its intensity. Multiply this equation by any arbitrary factor x and add 

 it to the preceding, and we have 



d_ } f -( 2 \ 



i A.J d, \ 6 1 



Regarding &, ... /3 6 as variables, the first member of this equation 

 equated to zero would denote a certain screw system of the second degree. 

 If that system were &quot;central&quot; it would possess a certain screw to which 

 the polars of all other screws would be reciprocal, and its discriminant 

 would vanish ; but the screw $ being absolutely the same as p, it is plain 

 that the discriminant of the second side must in such case also vanish. We 

 thus see that the ratios of the coefficients of the various powers of x in the 

 following well-known form of determinant must remain unchanged when 

 one co-reciprocal set of screws is exchanged for another. In writing the 



d z f 

 determinant we put 12 for -= , &c. 



16 1=0. 



26 



36 



46 



56 



66 + ape 



Take for instance the coefficient of of divided by that of x, which is 

 easily seen to be 



1 d 2 / I d 2 f 



PI da.1 2 p 6 da s 2 



* Williamson, Differential Calculus, p. 412. 



