668 Proceedings of the Royal Irish Academy. 



The emanants of the second degree are represented by the equation 



r. d 



'^■'v ^ f d 



d V„ 



do-i 



■*^'iK]^' 



"syhen F is the fimction into which / becomes transformed when the 

 co-ordinates are changed from one set of screws of reference to another. 

 If we take for /either of the functions already considered, these equa- 

 tions 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 very ingenious reasoning 

 employed by Professor Burnside in the theory of orthogonal transfor- 

 mations (see AYilliamson's Bijferential Calcidus, p. 412). 



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

 reciprocal screws of reference to another system. Let ^1 ... ^s he the 

 pitches of the first set, and qi, ... q^ be those of the second set. Then 

 we must hare 



Pifii^ + . . . + 2hfi6~ = SiH-i- + • . • + qelJ-^', 



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 



^^^^ • • • ^f^dXe]^'-''^^'^-'^ ■ ■ ■ -'^^^^'^- 



Regarding /3i, . . . ,8e as yaiiables, the first member of this equation 

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

 degree. If that system were " central" 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 /3 being absolutely the 

 same as /a, 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 determinant must remain 

 unchanged when one co-reciprocal set of screws is exchanged for 



= 0. 



cri . -i_u \> 



X J.LJJJ.^ LJ-LU '-IV^I.A^J. i-U.JJJ.ClJJ.U \\ 



V, JJL 



dai . doo 



11 + xpi, 



12 ,13 , 14 



) 



15 , 16 



21 



22 + xpo, 23 ,24 



> 



25 , 26 



31 



32 , 33 + xpi, 34 



) 



35 , 36 



41 



42 ,43 , 44-f 



xpi, 



45 , 46 



51 



52 ,53 ,54 





55 + xpi, 56 



61 



62 ,63 , 64 





65 , 66 + xps 



