Yector Differentials. 195 



of the unit-normal is the pure strain i/r-f \Jr'. Thus the 

 equation is of the first order with regard to Wv. 



8. In the paper referred to in Art. 1 it was proved that if 

 P be a scalar such that V 2 P = the unit-normal to the equi- 

 potential surfaces satisfies the equation 



VV(Vv.v)=0, (21) 



of which various expansions were given. If v be given 

 satisfying this condition P is determined by the equation 



logTVP = V" 1 (V^.v) J . . . (22) 



of which the solution is very direct and obvious. We may 

 thus write, as a set of equations defining orthogonal isothermal 

 surfaces 



Si/\/V=0 



S/a«r + ^)(2%-^ 2 )/^ + 2SA%^V 2 Vi/-0 . 



W(Vv.f)=0 



logTVP = V _1 (Vv.v) 



where the first two equations are to be satisfied by one unit- 

 vector in order that there may be three orthogonal families 

 of surfaces, the third equation must be satisfied by each of 

 the three unit-normals in order that these surfaces may all 

 be isotherms, and the last equation serves to determine the 

 three potentials. Cf. § 336 of Tait's ' Quaternions.' 



9. In studying special cases we have evidently at our 

 disposal a great variety of methods. Equations like (19) 

 appear to be chiefly useful in general investigations. In 

 testing whether any given family of surfaces satisfies the 

 condition discussed in Art. 7 it will usually be easier to find 

 a vector corresponding to one of the non-vanishing roots of 

 the strain-cubic, say parallel to rj, and operate on it with 

 S.77V, — though indeed the nature of the surfaces may be such 

 that (19#) takes a very simple form. As a brief example, 

 let a family of rings be denoted by the scalar function 



P=T^S- 1 / o^ J 



where q = ix +jy + kz + a and $>p — — {iic -\-jif) . Then 



dTq=-T- 1 q$pdp, 



so that by differentiating the given function, 



d¥= -4T 2 (/ S-V><fo> 8 P dp-2T 4 q S- 2 p</>p $dpcf>p. 



and because dF=—$dp\/F,> 



VP = 4pT 2 y 8- 1 / o# + 2^T 4 ^ S-7><fr>. 

 2 



