22 Prof. H. J. S. Smitli on the Conditions of 



not a sufficient condition for the existence of a pair of perpen- 

 dicular lines or planes. We proceed, therefore, very briefly 

 to describe the principal cases Avhich present themselves when 

 the coefficients are connected by one, two, three, four, or five 

 linear relations. By a linear relation connecting the coeffi- 

 cients we understand a linear homogeneous equation of the 

 type 



jyA + qB + rC + 2p'A' + %/W + 2r'Q' = 0, 



where j9, g, ?",/>^,^^, / are integral numbers which we may 

 suppose free from any common divisor. In connexion with 

 such a relation we shall have to consider the quadratic form 



^==j9P + qrl" + r?2 + ^p'r^ + 2r/f + 2r^^rj 

 and its contravariant or reciprocal form 



^ = (j/^-q,:),P + (<f-rp)/- + (y-pq)£' + ^pp'-<f^J)yz 



+ 2{c[(f — r'p'^zx + 2(r/ —p^q')xy. 



These we shall term the quadratic form and the reciprocal 

 quadratic form appertaining to the given relation. For brevity 

 we shall attend only to the cases in which given relations exist 

 between the six covariant coefficients A, B, C, A^, B^, C^, the 

 cases in which given relations exist between the six contrava- 

 riant conditions being simply the correlatives of these. It is 

 remarkable that in every case the conditions of perpendicularity 

 and symmetry depend solely on the coefficients of the linear 

 relations connecting the crystallographic coefficients ; so that 

 two parallelepipedal systems, in which the crystallographic 

 coefficients have different ratios but satisfy the same linear 

 relations, would resemble one another exactly in respect of 

 symmetry and perpendicularity. 



8. Case of one linear relation between the coefficients. 



Here we have the theorem, '^ The system contains a single 

 pair of perpendicular lines, or contains no such pair whatever, 

 according as the reciprocal form appertaining to the given 

 relation is or is not a perfect square." 



For the condition that the reciprocal form ^ should be a 

 perfect square, we may if we please substitute the condition 

 that the quadratic form -vfr appertaining to the given relation 

 should resolve itself into tv/o rational factors. Or, again, we 

 may replace this condition by the two conditions, (1) that the 

 discriminant of -^ is to be zero, (2) that the greatest common 

 divisor of the first minors of this discriminant is to be a per- 

 fect square. 



9. Case of two linear relations between the coefficients. 



We represent the quadratic forms and the reciprocal qua- 

 dratic forms apperfahiing to these relations by '^I'^'^i, '^2^''^^; 



