318 Proceedings of the Royal Society 
physical interest, inasmuch as they include the problem of finding 
two homogeneous strains, such that the vector-sum of their effects 
on any vector shall represent the effect of one given strain on that 
vector, while the effect of their successive performance in a given 
order on any vector shall be equivalent to that of another given 
strain. It is curious to compare this with the physical meaning of 
the differential equation from which these forms are derived. 
If g be one of the roots of the symbolical cubic in x (of which 
two will in this case generally be imaginary) and rj the correspond- 
ing unit vector, such that we have three conditions of the type 
(x “ 9)v = 
we have 
(g 2 - g<p + <10 rj = 0 . 
The vectors, which satisfy this and the two similar equations, are 
(all three) sides (real or imaginary) of the cone of the third order 
S .p<pp$'p = 0 . 
One curious result, which is easily derived from the equations 
above, is that, if a solid experience a pure strain, the planes in which 
any three, originally rectangular, vectors are displaced intersect in 
one line, 
4. On some Quaternion Integrals. By ProfessoPTait. 
(Abstract.) 
In my paper on “Green's and other allied theorems ” (Trans. 
R. S. E. 1869-70), I showed that 
f?dp =ffds V.UvVP, 
where P is any scalar function of p, and the single integral is ex- 
tended round any closed curve, while the double integral extends 
over any surface bounded by the curve, v being its normal vector. 
Writing 
a" = i'P -f- jQ + 
this gives at once 
fcrdp = ffds (S . UvV<^ - Y . (Y . UvV) cr) , 
of which the scalar and vector parts respectively were, in the paper 
referred to, shown to be equal. 
