428 Mr DAVIES on the Equations of Loci, Sfc. 



terpretation without actual reference to his book. His plan appears to be, however, 

 so far as I can infer it from certain passages in CUELLE, to take the equation of a 

 plane curve, and convert it into spherical form by writing sin or tan before the va- 

 riables, according to some law determined in his work, but which cannot be distinct- 

 ly gathered from his papers. 



If this be not his plan, his papers are perfectly unintelligible to me. If it be 

 his plan, it has not one feature in common with my own, except the simple funda- 

 mental idea of employing two superficial co-ordinates, instead of three linear ones, in 

 space. I will say more. If his method is to transform the plane equations into spheri- 

 cal ones, by the insertion of the trigonometrical functional characteristics amongst 

 the terms of the equation, it will certainly lead him into error. Such a principle can 

 only be founded on projective considerations : but I have ascertained that projective 

 considerations are insufficient even to obtain with certainty the plane from the sphe- 

 rical property ; and much more must they be insufficient to enable us to assign the 

 spherical from a knowledge of the plane equation. 



I should be doing great injustice to Professor GUDERMANN were I not to speak 

 in the highest terms of praise of the elegant theorems which he has enunciated, and 

 to which he has affixed his own methods of proof. I have gone through the greater 

 part of them by means of the system developed in these papers, and find them true. 

 Such a considerable number of properties of spherical figures, whilst they are cal- 

 culated to create a high opinion of the author's taste and skill, are also well adapted 

 to interest the attention of mathematicians by their beauty, both of algebraical form 

 and geometrical enunciation. 



ERRATUM, Page 388, line 11. from bottom, for p. 35, read p. 293. 



