66 Mr DAVIES on the Equations of Loci 



a specimen, had any thing of originality in it. Subsequent in- 

 quiries have convinced me that it has not been developed. In 

 the latter part of the summer, when reading the proofs of my 

 paper (which recalled my attention to the subject), I resolved 

 forthwith to undertake a complete analysis of its first principles, 

 and ascertain whether there were any insuperable difficulties in 

 the details, which deterred former mathematicians from the pur- 

 suit. I did so, and can now see pretty clearly the source of its 

 general neglect. The equation of a curve, when obtained, was not 

 commonly capable of ready and obvious interpretation, any symme- 

 try of which it was capable rarely extending to a symmetrical func- 

 tion of both variables ; those equations did not admit of classifica- 

 tion in any way analogous to those of rectilinear co-ordinates ; the 

 processes themselves were of a class very different from those with 

 which the majority of mathematicians were most familiar ; and the 

 tendencies of the transformations, often of considerable extent, exceed- 

 ingly difficult to foresee. Few, therefore, who had not had the 

 good fortune to be successful in some interesting collateral in- 

 vestigations, would have had the resolution to proceed with an 

 attempt so apparently endless and hopeless as to systematise the 

 mass of results that flowed from these preliminary, and in 

 some degree conjectural, inquiries. This last advantage I did 

 possess, and I hope the results given in the present disserta- 

 tion will show that it aifected me properly. The road is 

 now fairly opened, and I lose no time in pointing it out to ma- 

 thematicians having leisure and inclination for such pursuits. 

 As a whole, however, this paper may be said to be incomplete ; 

 but that has arisen from want of room, as in all cases the re- 

 sults admit of the same degree of completeness as that given 

 to the problems actually discussed. I have, however, dwelt 

 rather largely upon the equations of the circle, both because 

 it stands the most prominent, and perhaps most interesting, 



