498 Proceedings of the Poyal Society 
the armature seems at variance with what I have stated before, that 
less instead of more driving resistance is felt in closing either of 
the armature circuits, for here the new current seems to be paid for 
out of the motion of the armature. The discrepancy may possibly be 
accounted for by the consideration that both coils are now antago¬ 
nistic in their action, and that whatever part of the induced current 
appears in the magnetic coil, from whatever source derived, goes 
directly to oppose the conditions favourable to motion, and that 
between the opposing actions more heating in the core may be the 
accompaniment or equivalent of slower motion. When the coil of 
the electro-magnet is joined with the larger (electric) coil, so that a 
wire has only to touch the unconnected binding screw to close the 
circuit, and when the arm puts the machine into rapid motion, it is 
brought to an instant, one might say an impotent halt, on the wire 
touching the binding screw. One cannot help thinking, in trying 
such an experiment, that coil-brakes or drags may be yet extensively 
used in machinery. 
Whether this machine be any improvement or even a rival to 
existing machines, I do not pretend to say. I only wish in this paper 
to bring the peculiarities of its action before the notice of the 
Society. 
4. Mathematical Notes. By Professor Tait. 
1. On a Property of Self-Conjugate Linear and Vector Functions. 
In the course of an investigation connected with the free rota¬ 
tion of a rigid body I was led to the remark that, if £ and r\ be two 
vectors related to one another so that 
£ — V.r]<prj , 
where ® is a self-conjugate linear and vector function, we have 
also 
7] = Y.£<p£ , 
(so that the relation is reciprocal) provided 
S .r)Qr)'£ 2 r) = 1 , 
which implies also the corresponding equation 
S.£<p£f£ = l . 
