538 
Proceedings of the Royal Society 
(33) 
<fd' = ia = <t> 0 i ' Q , 
4>f =3 b = $ 0/0 > 
1 4*3 = hab — o4*o3 o ’ 
g)k — o — . 
Comparing (<f> o<f>o)i' o = witli ft 0 (<J) Q i' 0 ) = afi 0 i , &c., we get 
the corresponding relations 
^ = ^ o a > ^ o.v' • 
But generally we have 
flu) = ^Sato =|9(^/? 0 Saw)^ =p(4>’o 0) W • 
Hence forming i’ , f , by applying the operator _p( )q to i' 0 , j' Q (as 
it has already (2) been applied to k f 0 for obtaining k'), we get 
4>i = i'a , 
<«=/&, 
! fiifij = k'ab , 
<}!/& = 0 . 
Here the unit vectors i , j , k, represent directions which are 
invariably fixed, whereas i' , / , k' rotate with the forces /3 , ft , &c. 
(34) 
§ 6 . 
We introduce now 
p = ix +jy + kz , 
x, y , z, being the Carthesian coordinates of the point p on the 
single resultant of the system of forces, these coordinates being 
referred to axes which are fixed in space, and passing through the 
central point of the system. 
We have also, by introducing (33) into the expression (23) of 
4>4>'p, 
^ 4>4>'p = = “ icdSip =-jb 2 Sjp 
l = ia 2 x +jb 2 y . 
By this the equations (26) and (30) become 
( b 2 x 2 + a 2 y 2 + M 2 z 2 + M x - h 2 1 
( - 2 gabz j 
(35) 
( 36 ) 
