TRANSACTIONS OF SECTION A, 



633 



Hence the following simple vule : Make the equation to a curve in the x, y 

 plane homogeneous by the usual z ; interohange z and x, and we obtain the equation 

 to the projection. 



Thus the cubical parabola y"' = x = .r::- becomes Y'' = ZX^ = X'^, the semi- 

 cubical parabola. The same curve in a different position, viz., y = x'^, or yz- = x', 

 projects into YX- = 1 . 



In practice it will not be necessary to retain the two sets of letters ; we may 

 say that 



X I 



y = e--' = ~e' projects into y = xe^ 



V ■ ■ 1 



y = log .V = z log -- projects into y = .r log — 



Z .(." 



y = cos h.v = z cos h — projects into y =■ .i- cos h . 

 t/ = a,^ + ftj.r + a„.v- + a.,.r' + &c. 

 = rf(,s + a^.v + a.^'~+ a^—^ + . . , 



projects into 

 Or, generalising, 



y = a^.v + ai + «,- + c^- + 



y = /(.t) projects into y = .i/^- j . 

 From the transformation 



^ = X'" = X 



we obtain 



dy _ Y — X'^^ 



d.v 



dy dY 

 '' dx dX 



dX 



Hence the solutions of 



project into those of 

 Also 



therefore 



•^ dx •' \dx) 



Y-X^=f'i^\ 

 dX •' \dXj 



^i!y = x^ f^ 



dx- " dX' 



'P d^Y 

 " f dXr 



,d^y ^ y.cPY 

 ^ dx' dX' 



If the constants z, Z be retained, the relation is 



y' dy^Y^ d^ 

 s^ dx- Z- dX^' 



of ?ero dimensions, as might be expected. 



