72 Proceedings of the Royal Irish Academy. 



may take as origin tlie, unique point which returns to itself, and thus the 



constants a, b,c, d . . . disappear. That there is snch a point follows from 



solving the equations 



X = a + l-^x + m-^y + . . . 



y = h + InX + tihy + . . 



z = c + 49; + . . . 



The determinant /(I) does not vanish in general* Hence x, y, z, . . . 

 may be found uniquely. 



22. The central ccxis when n is odd. — In § 19 we have taken the origin on 

 the central axis. In general no point returns to itself, since the determinant 



/(I) vanishes (§ 19). But we can directly prove the existence of the 

 central axis and find its equation. It is a line such that every point 

 thereon returns to the line. Now, suppose that (P), x, y, z, . . ., 

 (A), a, /3, 7, . . . are two points on such a line, and that its direction- 

 cosines are L, M, N, . . . Then 



X = a + L9, y = fi + MO, etc. 

 Now the displacement of P must be equal to the diplaeement of A, and 

 we have therefore n equations of the form 



(l, - l)x + m^j + 7ij« + . . . = (Zj - 1) ft + mj/3 + Wi7 + . . ., 



and the conditions are satisfied if Z, M, N, P, . . . are proportional to the 



minors formed by omitting any row of /(I). The equation of the central 



axis is then 



a + (/j - 1) a + mj/3 + n-^y + _ I + Ua + {m^ - 1) j3 + n^y + 



— ^ - -^ - etc. 



and each of those equals the invariant of translation. 



23. The invariant of translation when n is odd.f This is the quantity 

 which remains unaltered when the axes are translated parallel to themselves, 



and is 



t = Za + 3Ih + M + Pd+ . . . 



The transformation consists first in choosing any origin on the central 

 axis, when the equations are 



x' = Zt + IjX + in{y -t n-^z + . . . 

 y' = Mt + I2X + m^y + n^z + . . . 

 sf = Nt + l^ + m^y ¥ rhz + . . . 

 etc. 



* See note, p. 70. 



t This invariant does not exist when n is even, sincR / (1) does not vanish in the 

 general case. 



