176 Professor Baker, On the transformation of the equations of 



en = 



which give for instance 



— eiVag = {hw - gv) M^^ + {hw -fv) M^^ + {fw - cv) M^^ 



+ {he -f')M^,+ {fg- ch) M,, + {hf-bg) M^, 



— £^^14 = {ad — u^) M^s + {hd — uv) M^^ + {gd — wu) M^^ 



+ {hw — gv) ilfi4 + {gu — aw) M^ + {av — hu) M^, 



— eiVi2 = (O^'y — ^^**) -^23 + (^^ ~" ^^) -^31 + (5'^ — /^) ^12 



+ {hf - bg) Mi4 + {gh - af ) M^ + {ab - h^) M^. 



The reader will compare the formulae for the polar hne of a line 

 in regard to a quadric, Salmon's Solid Geometry (1882, p. 60), and 

 may verify that, the coordinates of a Hne being I, m, n, V , m' , n' , 

 where the Hne is given by I't + mz — ny = 0, I't + rn'y + n'z = 0, 

 the polar Hne (A/>tvA'//,V') of {Imnl'm'n') in regard to the quadric 

 {a, b, c, d, f, g, h, u, v, w\ x, y, z, t)^ = is given by 



M,„ Mo 



0, 



n', 



m' 



I, 



^34 



3l5 -'^12' 



-n', 

 0, 

 I', 



m' , I 

 V , m 

 0, n 



— m, — n, 







were proportional to 



I, m, n, - V , — m' , — n' , then N^z, N^^, N-^^, TV^^, iV24, iV34 would 

 be proportional to A, /jl, v, — A', — /a', — v' . We assume however 

 that M^^M-^^ + M^-^M^i + M^^M^, and the corresponding ex- 

 pression for Nij, are not zero. The above expression of n in terms 

 of m gives, on taking the determinants of both sides, eV = /x^S^ />t~^, 

 or, from e* = 8^, v^ = y?. The expHcit forms for N ^j show that t- =.— /a. 

 We consider now a transformation, from the variables {x, y, z, t), 

 upon which a, h, g, ..., M^j, N^j, depend to variables {x', y', z', t'); 

 we write 



dx = j)-j^idx'+ Pi2dy'+ Pi^dz'^ Pi^dt', 



dy = P2^dx'+ j322#'+ P23<^^'+ P2idt', 



and so on, or more briefly, with the ordinary matrix notation, 



{dx, dy, dz, dt) = p {dx', dy', dz', dt'), 

 the converse equations being written 



{dx' , dy' , dz' , dt') = p' {dx, dy, dz, dt). 



