222 ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION [32 
: Vi ■ z l = */{W(e, e, q)} x (90), 
{W(e, p, q) W (eV[, eY\ q) + W(e', p, q) W( eV[, A q) + TF (e", p, q) TF ( eV[, ee\ q)} 
: V{TT(e', e', q)} x 
{W (e, p, q) W ( 7e, eV[, q) + TF (e', p, q) TF ( ej?, e!\ q) + TF (e", p, q) TF (e"e L ee\ q)} 
: V{W(*", e", q)} x 
{W(e, p, q) W(e7, 77, q) + TF(e', p, q) W(e7, Te, q) + W(e", p, q) W (77, 77, q)}; 
these may be reduced to the very simple form 
: Vi ■ z 1 = J{W(e , e , q)} TF( e'e', w, q) •• (91), 
: 7{W(7, e', q)} W (77, ®, q), 
: V{Tf (*", q)} «, q). 
and in like manner we obtain 
: 2/i : *1 = \/i^( e , e , q)} F ( eV, q) (92), 
: </{W(7, e', q)} W(7e, p, q), 
: 7{W(e", 7, q)} TF@ p, q). 
It will be as well to indicate the steps of this reduction. Consider the quantity 
in { } in the first line of the equation which gives the ratios x x : y 1 : z 1 \ and suppose 
for a moment e'e” = l, m, n, &c.: then, selecting the portion of the expression which is 
multiplied by a, this is 
= al {l (af + b v + cQ + V (a'f + b'v + c'g) +l” (a"f + b" v + c"£)}, 
la + IV + l" a" = 777, Z6 + ¿'6' + l"b" = 0, Ic + He' + l"c" = 0, 
this reduces itself to ee'e". a££, which is a term of 
ee'e" TF(e'e", «, q); 
and by comparing the remaining terms in the same manner, it would be seen that 
the whole reduces itself to 
777 W( 77, ft), q); 
whence the formulae in question. 
The formulae (86), (87), (91), (92), completely resolve the problem of the transfor 
mation of coordinates; they determine respectively p x from p or w, w x from p or <y. 
To complete the present part of the subject we may add the following formulae. 
Suppose : y x : = ax x + o!y 1 + a"^ (93), 
: hx l + b'y 1 + b'b/i, 
: ca?! + c'y x + q" Z\,