G. PLARR ON THE SOLUTION OF THE EQUATION Vptp=0 : 89 
These expressions of @ do not vanish for any point of the surface [=0, 
save and except the two first, for the particular point 
e=¢y1, Where ¢€=(h—he)(he—h);, 
because then we have W=/h,, and as generally 
we=Be= —DhySre , 
we get for the above point : 
we = Ch p=he , 
So that (a—W)e=0, and @1, ©, vanish, and consequently y, and woo 
give the result zero for w of any direction whatever. 
But in this case the direct treatment of Veap=0, with the hypothesis 
ap=—Bp+éVpup, gives for p the two following solutions :— 
p=(—A+vr), and p=(—A7 + v)z + py, 

where 7 = — ; and y being independent from one another. 
The first solution is determinate, and, according to (65) of § 11, it repre- 
sents the normal to the plane R; the second is indeterminate in so far as it 
represents by (66) § 11 any direction parallel to the plane R’ (not R). 
__ When on the curves characterised by P=0, Q=0, the value of W vanishes, 
and the cubic, /a=0, reduces itself to a =0, it does not follow that o=0, 
and neither will Tae vanish , nor €,, @,, will, nor @;; the only remark to be 
made is: that the cubic, 7a=0, is founded on the supposition that Q does 
not vanish, and, when this circumstance takes place, the cubic is to be replaced 
by the equation from which it was originally deduced, namely, from p=a’, in 
this particular case of P=0, Q=0. 
And yet, if the cubic 4m = 0 fails in this instance, there exists at any rate 
a corresponding scalar equation in « (see § 9, (51 ter.) ), namely, 
Sea'e — 0, 
which is the resultant of P= 0, @=0, and in which any tensor may be. 
reintroduced as a factor of the unit vector ¢ = Ue. 
VOL, XXVIII. PARTI. - Z 


