96 Prof. Young on the Theory of Conjugate Points. 



point: the imaginary curve, according to the inference of Mr. 

 Gregory, " has contact of the second order with the plane of 

 the axes*." 



I have employed the term imagmary in the foregoing ob- 

 servations in accordance with the usual phraseology ; but the 

 views of Mr. Warren and Dr. Peacock, in reference to the 

 geometrical interpretation of the roots of unity — and which 

 views Mr. Gregory and Mr. Walton have done so much to 

 illustrate and confirm — will necessarily lead to the abandon- 

 ment of this expression in the application of analysis to curvesf. 



From what has now been shown, it appears that it is not 

 " useless to attempt a test of a conjugate point by the differ- 

 ential calculus:" since this point has tlie same situation as 

 another which unites with the continuous series of points form- 

 ing an imaginary curve, or a curve out of the plane of the co- 

 ordinates, and to any continuous series of values the calculus 

 is, of course, always applicable. But we may connect the 

 isolated point with a continuous series of real values, by re- 

 garding it as the extreme case of a series of diminishing ovals. 

 By confining our attention to this latter continuous series, the 



d V 

 values of -r^ at the point become innumerable, and all real ; 



by confining our attention to the former continuous series, the 



(I V 



values of -T^ become imaginary; but if we consider the point 



in its detached state, unconnected with either series, then, of 

 course, to attempt to apply the calculus would be absurd. 

 The equation 



[x - af + (3/ - Z>)2 = 



* In strictness it has contact of the second order with a certain curve in 



this plane J the curve, namely, of which every point, indifferently, is repre- 

 ss 

 sented by the co-ordinates of the conjugate point, x-=.a,y=.-^i when a is 



supposed to vary J in other words, the curve whose equation is5^=-j; for 



y, -^ and — ^, when a is put for x, will, with the exception of the imagi- 



nary zero, be the same in both curves. When the imaginary curve is 

 viewed as a real curve out of the plane of co-ordinates, as Mr. Gregory 

 views it, I must confess that I do not see the grounds of the inference, 

 quoted above, as to the order of contact with this plane. 



f The writer of these remarks is anxious here to express his conversion 

 to these views of Dr. Peacock ; and he wishes to do so thus publicly, be- 

 cause upon a former occasion (Mathematical Dissertations, p. 30) he ven- 

 tured to state his misgivings as to their accuracy, although he took care to 

 do so with that becoming hesitation and respect which he felt to be due 

 towards so high an authority. 



