556 



boloid, this simple and graphic process can actually be applied, 

 without any such modification from imaginaries as was above 

 alluded to. The consideration of non-central surfaces does 

 not enter into the object of the present communication; nor 

 has it been thought necessary to consider in it any limiting 

 or exceptional cases, such as those where certain positions or 

 directions become indeterminate, by some peculiar combina- 

 tions of the data, while yet they are in general definitely as- 

 signable, by the processes already explained. 



IG. Sir William Rowan Hamilton is unwilling to add to 

 the length of this communication by any historical references; 

 in regard to which, indeed, he does not consider himself pre- 

 pared to furnish anything important, as supplementary to what 

 seems to be pretty generally known, by those who feel an in- 

 terest in such matters. He has however taken some pains to 

 inquire, from a few geometrical friends, whether it is likely 

 that he has been anticipated in his results respecting the in- 

 scription of gauche polygons in surfaces of the second order; 

 and he has not hitherto been able to learn that any such an- 

 ticipation is thought to exist. Of course he knows that he 

 must, consciously and unconsciously, be in many ways in- 

 debted to his scientific contemporaries, for their instructions and 

 suggestions on these and on other subjects ; and also to his 

 acquaintance, imperfect as it may be, with what has been done 

 in earlier times. But he conceives that he only does justice 

 to the yet infant Method of Quaternions (communicated to the 

 Royal Irish Academy for the first time in 1843), when he 

 states that he considers himself to owe, to that new method 

 of geometrical research, not merely the results stated to the 

 Academy in the summer of 1849, respecting these inscriptions 

 of gauche polygons, and several other connected although 

 hitherto unpublished results, which to him appear remarkable, 

 but also the suggestion of the mode oi geometrical investiga- 

 tion which has been employed in the present Abstract. No 

 doubt the principles used in it have all been very elementary, 



