of Edinburgh, Session 1862 - 63 . 117 
terms of the form Yep in the expression for <r', which will then 
express a distortion accompanied by rotation. 
Also, a solution of <^q = a (where q and a are quaternions) is 
q = SCp + Yep + cf3p. 
It may be remarked also, as of considerable importance in phy- 
sical applications, that, by (8) and (9), <i (S-1- JV)ap = 0, but I 
cannot enter at present into details on this point. 
IV. In this brief note, I shall not give any more of these trans- 
formations, which really present no difficulty; but I shall show 
the ready applicability to physical questions of one or two of those 
already obtained, a property of great importance, as it may now 
be asserted that the next grand extensions of mathematical physics 
will, in all likelihood, be furnished by quaternions. 
Thus, if cT' be the vector-displacement of that point of a homo- 
geneous elastic solid whose vector is p, we have, p being the con- 
sequent pressure produced, 
<I <1 ^ cr- =0 (16) 
whence S8p<l ^a-'= — SSp <\p = Sqy, a complete differential.. . . (16)^ 
Also, generally, ^ = A:S <] a^, 
and if the solid be incompressible 
S<1<^ = 0 (17) 
Thomson has shown (fjamh. & Dub. Math. Journal, ii. p. 62), that 
the forces produced by given distributions of matter, electricity, 
magnetism, or galvanic currents, can be represented at every point 
by displacements of such a solid producible by external forces. It 
may be useful to give his analysis, with some additions, in a 
quaternion form, to show the insight gained by the simplicity of 
the present method. 
Thus, if S(3-8p = si, we may write each equal to-S5p<j~. 
Ip Ip 
This gives 
the vector-force exerted by one particle of matter or free electricity 
on another. This value of evidently satisfies (16)^ and (17). 
VOL. v. 
p 
