182 

 then 



Prof. Ca'jley, On the transformation [Oct. 28, 



gi. fi' c, 

 and the second equation written in the form 



H^a,, h,, gjr=/.A 



in fact denotes the six equations 



(a,, b,, c,, f„ gj, hja, a', a")' = Z; , 



(/3, /3', /3")^ =/!:, , 



(7 , 7 > 7'T = I^i > 



i^, 0, /3'l7, 7'> 7")=/^A» 



(7. 7', 7"]I« . ci' . a") =-^1^1, 



the two sets each of six equations being in fact equivalent to a 

 single set of six equations, and serving to express the relations 

 between the nine cosines (a, /3, 7, a, /3', 7', a", /S", 7"), and the 

 cosines (\, //-, i^) and (\^, /x^, vj. Observe that the nine cosines 

 are not (as in the rectangular transformation) the coefficients of 

 transformation between the two sets of coordinates. 



From the original linear relations between the coordinates, 

 multiplying the equations of the first set by x, y, z and adding, 

 and again multiplying the equations of the second set by (x^, y^, z^ 

 and adding, we have 



{Q.\x ,y ,zY={ WJx^, 2/1. Zi\^ ^y >^ ), 



But (T^fe, y„ z^J^x, y, z) and {VJx, y, z\x^, tj^, z^ 

 denote one and the same function ; hence 



{p.\x, y, zf = {^J^x^, y^, z^)\ 



that is, 



(1, 1, 1, \ IX, vjx, y, zf = (1, 1, 1, Xj, /A^, v^x^, y^, z^\ 



