78 
MACFARLANE. 
The next step is to find pr where r is the modulus 
l/V 4- y 2 + 2 2 - Now, 
1 ___ * 1 1 _ 2/ 1 ^ 1 „ z 1 
i r i ' 8y j r j ’ dz k r k \ 
Therefore 
Here p denotes the spherical axis of the vector R, r denoting 
its modulus. 
According to Professor Tait’s principles, pr — p; but this 
result involves the assumption that i -^4 is equivalent, for 
real vectors, to 4- This may do, so long as the axis is 
constant, but when it is variable the nature of its dimensions 
cannot be changed arbitrarily. If its dimension is — 1, it 
cannot be -f 1; p has one dimension in length and 4- has 
minus one dimension in length. It will be found that 
pr = -4 is consistent with, while pr = /> is inconsistent with, 
the further development of the method. 
The next step is to find pp. We have pR — 8 ; but 
pR — (7 (rp) — pr . p -f- rpp ; therefore 
2 
3=1+ rpp ; therefore pp — —. 
r 
The next point is to show that pR i is not the same thing 
as pr 2 , for 
pR 2 = p (r 2 p' 1 ) = 2 rpr . p 2 + 2 r 2 p pp = 6 rp ; whereas 
2 r 
pr 2 = 2 rpr — —. 
P 
Here there is a difference both in the numerical multi¬ 
pliers and in the dimensions of p. The principle of Hamil- 
