20 Prof. H. J. S. Smith on the Conditions of 



fessor W. H. Miller, we designate by a, h, c the parameters 

 appertaining to the three lines of the system taken for the co- 

 ordinate axes ; we also denote by X, Y, Z the angles between 

 the coordinate axes, and by Xi, Yj, Zj the angles between the 

 normals to the coordinate planes. We thus have for the 

 square of the distance between any two points of the system 

 the expression 



f(x, 7/, z) = a V + 5y + cV + "2hcyz cos X + 2cazx cos Y 



+ '^ahxy cos Z, 



where x^ ?/, z denote any positive or negative integral num- 

 bers ; and this ternary quadratic form may be regarded as cha- 

 racterizing the given parallelepipedal system. Again, if 



^(f , 77, = h\^ie sin'^ X + 6'aW sin^ Y + a%^\^' sin^ Z 



+ ^a^bcT]^ sin Y sin Z cos X^ + 2b'^ca^^ sin Z sin X cos Yi 



+ 2c^ab^r) sin X sin Y cos Zi, 



the form (/>, which is the contravariant of/, characterizes (in 

 the same way in which / characterizes the given system) a 

 new parallelepipedal system (the polar system of Auguste 

 Bravais) in which every line is perpendicular to a plane of the 

 given system, and in which the parameter corresponding to 

 any line is the elementary parallelogram of the given system 

 lying in the plane to which the line is perpendicular. 

 6. We write for brevity 



/= Ax^ + Bzf + Gz'' + 2A'i/z + 2B'xz + 20' xy, 



0=Aif + Bi7;2 + Ci?2 + 2A\9?? + 2B^fr+2C^f7; 



(so that A = a^, . . . A' = &c cos X, . . . , Ai = h^c^ sin^ X, . . . 

 A.\ — c^hc sin Y sin Z cos Xi, . . . ) ; and we observe that, although 

 the five quantities upon which the nature of the parallelepipedal 

 system ultimately depends are the ratios of the parameters 

 a, b, c, and the three angles X, Y, Z, yet the combinations of 

 these quantities which it is most convenient to consider in 

 discussing the conditions of perpendicularity are precisely the 

 six coefficients 



A, B, C, A^, B^, C^ 



and the six contravariant coefficients 



Thus the condition that the lines of the system 

 x_ _y_ _ 3 



au bv cw 

 X _ y __ 3 

 aui bvi ctoi J 







