practical Instrument of Physical Research. 481 



potentials illustrated by applying it to a general electrical 

 problem. (2) Two examples in curvilinear coordinates. (3) A 

 quaternion proof of a well-known theorem of Jacobi's of great 

 utility in Physics. (4) A generalization of one of the well- 

 known integrals of fluid motion. (5) The well-known par- 

 ticular solution of the differential equation expressing the 

 conditions of equilibrium of an isotropic elastic solid subject 

 to arbitrary bodily forces. (6) A short criticism of Prof. 

 Poynting's theory of the transference of energy through an 

 electric held. 



In the proofs below, so far as quaternion knowledge is con- 

 cerned, an acquaintance with Tait's ' Quaternions,' 3rd edit., 

 only will be assumed. The following two equations from 

 §§ 498, 499 of that treatise will be frequently required below. 



$dpq = $VUv Vl qds, (1) 



§U Vq ds=$ V qd, (2) 



In equation (1) ds is an element of a surface, UV the unit 

 normal at ds, dp a vector element of the boundary, and q a 

 quaternion function of a point in space. In equation (2) cfc 

 is an element of volume, ds an element of the bounding sur- 

 face, Uv the unit normal at ds pointing away from the region 

 bounded. If the surface in (1) contain lines of discontinuity, 

 or the volume in (2) surfaces of discontinuity in q, the equa- 

 tions are still true if such lines and surfaces of discontinuity 

 are included in the boundary of the region. In such a case, 

 of course, the elements dp and ds will each occur twice in the 

 integrals (1) and (2) respectively, namely once for each of 

 the two regions bounded by the element. 



I. Potentials. 



In the volume and surface integrals that are now required 

 it is necessary to pay attention to the following convention. 

 Let p a be the vector coordinate of a certain point under con- 

 sideration, and let p b be the vector coordinate of the element 

 of volume t/?. It will frequently happen that we have to 



deal with integrals of the form (Tf (£(V) cfc, where <j> is any 



quaternion function of a quaternion r. [In all the applica- 

 tions below, (j> will be a linear function, but this is not 

 necessary.] The form of (f) is a function of p b only, and r is 



a function of p b —p a only. Thus in the expression Tj\\\ <p(r)(h 

 the only meaning that can be given to the differentiations 

 implied by y is such that these differentiations require p in 



