66 _ Proceedings oj the Royal Irish Academy. 



The orientation of )S„_2 is therefore (§ 3) 



I = liX + villi + . . . . 

 m = kX + m^ix + . . . . 

 etc. 



Then a rotation through an angle Q is expressed by 



x' = ^ + (x - £) cos ± ^ sin 6, 

 7/' = ,, + (7/ - i() cosO ± £ sin 0, 

 z' = Z + (z - L,) cos 6 ± C sin d. 

 v = w ^ (v - (.1) cos Q ± D sin d., 

 etc. 



■where 



and 



I = 2i""- l,{l,x + viry + n,-z + p,.v, + 

 1} = '2i"~-m,(l,-o: + vii-y + rii-z + prU + 

 etc., 



A = 



y 



■'«3 



z 



th 



n-. 



u 

 th 



V 



23 



■)- 



with similar forms for B, 0, B, etc.* 



This rotation formula may be verified almost as if n were equal to 3. Note 

 that (5, ij, f, w, . . . .) is the foot M of the (single) pei-peudicular from 

 P(x, y.z, , . . .), and if P' is tlie new position of P we have only to prove 

 that M is also the foot of the perpendicular from P', and 



F2r- = 1(x - |)= = 2(.b' - if = P'jr, 

 5.r = 2,r'- = ;•-, 



cos PjUP' 



— — = COS y. 



16. Co-ordinates of a complete system qfrotatioiis. — Let us call the system 

 i2„. Then P„ means a series of rotations round the flats of a "complete 

 system" (P„) of mutually plane-normal flats (§ 7). The independent 

 co-ordinates of P„ are therefore those of P„ together with the angles of 

 rotation. 



' The sign to be placed before A, B, etc., must be fixed by some arbitrary convention, 

 ■which need not delay us. 



