12 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



It is now easy to show by means of (16) that 



2 (bfe) = (?qr) L X M,N X 



(27) 2 (eg a) = (p q r) L 2 M 2 N 2 

 2 (ahb) = (ft qr) L 3 M 3 N 3 



and hence by (26) that 



& : ki : Z-3 = (bfc)- 1 : (eg a)- 1 : ( a h b )" J 



Substituting these values in (23) w<? /jtfZ'e the formula of 

 the inverse transformation in the following final form : 



P x, = (bfc)-* [(ybcy — i(ybf) (yfe)] 



(28) p x 2 = (eg a)~ l [(y c af — 4 (yeg) (yga)] 

 p x 3 = (a h b ) -1 [(_y « ^) 2 — 4 (yah) (y h b )] 



With these formulae the object of the present paper is 

 accomplished. The method will fail in special cases, but 

 in such cases the numerical relations between the coefficients 

 are always such as to make the inversion by other methods 

 comparatively easy. For instance, we see by (28) that the 

 pointy is the polar of the line b c with respect to <& = o. 

 Now if it happen that_/* lies on this line our formula for <E> 

 may vanish identically; but in this case evidently <£> degen- 

 erates into two lines of which b c is one. Again, the 

 method will sometimes fail when two of the points ft, q, r 

 coincide; but in this case the Jacobian has a square factor, 

 which can therefore readily be found, and the formulae re- 

 duced at once. 



Berkeley, California, 



October 18, 1897. 



