( 45° ) 
IV.—On the Solutions of the Equation Vp¢p=0, op representing a Linear 
‘Vector-Function, generally not Self-Conjugate. By Gustav PLarr, 
Docteur és-sciences. Communicated by Prof. Tarr. 
(Received July 26—Read December 18, 1876.) 
INTRODUCTION. 
Certain cinematical and physical questions lead to the problem: to determine 
the directions in which a given linear vector-function, ¢p, assumes a direction 
parallel to that of the vector, p, on which it depends. 
The condition of parallelism is expressed by 
Vpop=0, 
and it is translated into the equation 
where .g represents a certain scalar, on whose determination the whole 
problem depends. 
Following the method traced out by Hamitton, we treat this equation 
successively by . 
Sa( ), SBC), Sy(_), 
a, B, y, being any system of vectors not coplanar between each other ; but we 
will state at once that throughout the whole of this paper we shall assume 
a, 8, y, to form a system of ¢reble rectangular unit-vectors, of which hypothesis 
the justification is evident. 
Designating by ¢’ the conjugate of ¢, defined according to 
Sogp=Sp'c , 
we arrive at the known results (1st), that the vectors 
(¢—g)a, (@—9)B, (¥ 9) 
are all three in one and the same plane, are coplanar, in one word ; and (2d), 
that the sought-for direction of p, satisfying to Vp¢p=0, is perpendicular to 
that plane. 
The first of these results gives us the scalar equation 
(1) S(¢'—g)a(¢ —9)B (¢:—g)y=9, 
which expresses the coplanarity of the three vectors forming the product ; and 
VOL, XXVIII. PART I. M 

