180 Prof. Cayley, On the trajisformation [Oct. 28, 



Coming now to the question of transformation, write 



viz., the axes of ic, 3/, z are inclined to each other at angles the 

 cosines whereof are \, fjb,v: those of x^, y^, z^ are inclined to each 

 other at angles the cosines whereof are \, f^^, v^: and the cosines 

 of the inclinations of the two sets of axes to each other are a, /3, 7 ; 

 °-\/3iy'', a", /3", 7"; as is more clearly indicated in the diagram, 

 the top-line showing that 



cosine-inclinations of ic to oc, y, z, x^,y^,z^ 



are 1, v, fj,, a , a! , a" respectively, 



and the like for the other lines of the diagram. The letters 

 O, Ilj , V, W are used to denote matrices, viz., as appearing by the 

 diagram, these are 



( 1, V, fi), (1, v„ /ij, (a, 13, 7 ), (a, a', a") 



V, 1, X I'l, 1, \ a', 13' , 7' A /3', 13" 



/^, \ 1 /^,, \. 1 a", /3", 7" 7. 7'. 7" 



respectively. 



The coordinates {x, y, z) and (cCj, y^, z^ form each set a broken 

 line extending from the origin to the point ; hence projecting on 

 the axes of x, y, z and on those of x^, y^, z^ respectively, we have 

 two sets, each of three equations, which may be written 



(Si\x, y, z) = {W\x^, y„ z^, 



{V\x, y, z) = {D.Jx^, y^, z^) ; 



and where of course each set implies the other set. 



We have 



(x, y , z) = {n-'W^^x^, y„ ^,), = {V njx^, y^, z^), 



{x,, y„ z,) = ( W-'D.Jx ,y,z), = (11,- VJx,y,z), 



the first giving in two forms {x, y, z) as linear functions of 

 (^1) 2/i. ^i)> ^"^^ ^^® second giving in two forms {x^, y^, z^ as linear 



