94 Prof. J. R. Young on Imaginary/ Zeros, 



in the foot note at p. 71 of his first volume, as also at p. 268 

 of the second volume. 



It is a valuable peculiarity of algebra, that it supplies, by 

 indications of its own, cautionary information which, if at- 

 tended to, would prevent hasty interpretations of this kind. 



The symbol — -, as is well known, is often such an indication ; 



and, as Lagrange has observed, is the form which analysis 

 assumes to preclude a violation of its own laws. 



The hitherto neglected symbol v' — 1, is a symbol of like 

 character ; its office being to apprise us that whenever we ar- 

 rive at it, by passing over a continuous series of values for the 

 quantity that has thus become zero, the corresponding conti- 

 nuous series of values of the function itself is an imaginary 

 series, with which imaginary series the value arrived at neces- 

 sarily unites, and must thus be itself considered as imaginary. 

 It is possible that if this imaginary zero be arrived at through 

 a descending series of values, it may appear as a real zero 

 when reached through an ascending series, and vice versa. 

 But this implies no contradiction, nor would it lead to the 

 conclusion that the same point would, in such a case, be both 

 real and imaginary ; the proper inference would be that the 

 imaginary point is superposed upon the real point, or that the 

 imaginary branch of the curve passes through a real point of 

 the locus. If, however, the zero continue to be imaginary, in 

 whichever way we arrive at it, then the values of the function 

 continue to be imaginary after the passage through this zero, 

 as well as before. At the passage a real point no doubt exists 

 — not however to interrupt the continuous series of imaginary 

 values adverted to — it exists independently of that series, and 

 free from all connexion with it; inasmuch as the co-ordinates 

 of the point satisfy the equation though the symbol v^ — 1, 

 implying such connexion, be suppressed ; the co-ordinates of 

 this conjugate point will therefore be determined by finding the 

 co-ordinates of the imaginary point situated upon it, and then 

 suppressing the symbol \^ — \, 



In like manner there is no interruption of continuity in those 

 curves that have what have been caWed punctuated branches \ 

 the continuous branches, whether real or imaginary, exist in- 

 dependently of the assemblage of real points constituting the 

 punctuated branch ; and, as among these no continuity ever 

 existed, it would be wrong to speak of any interruption of con- 

 tinuity in reference to them. 



From the foregoing considerations it is easy to infer plain 

 and unequivocal rules for the determination of conjugate points, 



