§S Prof. J. R. Young on Imaginary Zeros, 



" As ^ it is thus important to inquire whether the sign of an 

 evanescent quantity be plus or minus, may it not be of equal 

 importance to ascei'tain, in certain investigations of analysis, 

 whether that sign be real or imaginary? That attention to 

 this is of consequence in the application of algebra to geometry 

 will appear by taking the extreme case of the ellipse, when by 

 the continual shortening of the minor diameter, the curve ulti- 

 mately degenerates into a finite straight line — the major dia- 

 meter: the equation to this ultimate form of the ellipse is 







y — — V c^ — x^. 

 ^ a 



Between the limits x=—a and x—+a, the evanescent 

 quantity on the right is real, implying that every point on the 

 axis of X, which does not lie without these limits, is a real 

 point of the locus. But if x exceed a, in either the positive 

 or the negative direction, the evanescent quantity is imaginarT/; 

 and consequently, in accordance with the ordinary interpre- 

 tation, the corresponding points on the- axis of x are beyond 

 the bounds of the locus. 



If the curve had been an hyperbola instead of an ellipse, 

 the equation, in the extreme case, would have represented two 

 infinite straight lines in directum with the axis of abscissas, 

 but separated from each other by an interval equal to the fixed 

 principal diameter. 



But it is in the theory of conjugate points that the imagi- 

 nary zero is in a more especial manner an item of importance ; 

 and it is from the disregard that has hitherto been paid to it, 

 that the discrepancies and contradictions, to be found in the 

 most recent expositions of that theory, chiefly arise- 

 That such contradictions really exist may be easily shown : 

 thus Mr. Walton, in a very ingenious and instructive paper 

 on this subject in the Cambridge Mathematical Journal, vol. 

 ii. p. 164, says, in reference to the doctrine almost universally 

 taught, " Several writers on the calculus have erroneously 



supposed -T^, at a conjugate point, is necessarily imaginary." 

 And the late distinguished Mr. Gregory, in his valuable Ex- 

 amples, states that " it may happen that ~ and any number 



of the differential coefficients are possible at a conjugate 



point" (p. 162). In the preceding page we find it affirmed 



d v 

 that, " If -f^ is found to have two or more equal and possible 



values, there are two or more branches of the curve touching 

 each other in one point, which is called a point of osculation." 



