and the Theory of Conjugate Points. 93 



And yet at page 170 it is said, " Tiie curve whose equation is 



{c^i/—x^)^= {x—hY'{x—a)% a<b ,. .- - . 



has a conjugate point whose co-ordinates are .r xsc, y s?: -*^, 



but the differential coefficients are possible till we come to the 



third." Now these differential coefficients, thus affirmed to 



be possible, give each a pair of equal values ; so that the point 



under consideration ought to be pronounced, not a conjugate 



point, but a point of osculation*. 



Again, the most prevalent doctrine respecting conjugate 



points is, that every such point may be correctly regarded as 



an evanescent oval; and simultaneously with this view it is 



also maintained that the existence of the point is indicated by 



d v 



— becoming imaginary {Lacroixy English translation, p. 99). 



But in Mr. Walton's paper, before referred to, it is affirmed 



that, according to the theory of the evanescent oval, "-y^ 



ax 



must remain perfectly indeterminate;" while Prof. De Morgan 

 declares (Differential and Integral Calculus, p. 382) that " it 

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

 ential calculus." 



It is clear, from the above contradictory views, that the 

 theory of conjugate points is still in an unsettled state; but 

 before attempting to reconcile these contradictions — for after 

 all they are more apparent than real — it may be well to offer 

 a remark or two on the principle of continuity in reference to 

 inquiries of this kind. !) 



It appears to me that this important principle universally 

 prevails in all the general expressions of analysis; and that it 

 is incorrect to affirm that it suffers any breach or interruption 

 in any instance whatever. No doubt a continuous series of 

 real values, furnished by a general analytical expression, may 

 terminate, and a new series — a series of imaginary values — be 

 originated, both series conforming equally to the law implied 

 in the general form; but no instance can be adduced in which 

 a continuous series of imaginary values becomes interrupted by 

 the interposition of a single real value, and is then again re- 

 sumed ; although such a remarkable interruption is frequently 

 said to happen at a conjugate point, and analogous circum- 

 stances are often said to occur in other analytical inquiries, as 

 for instance by Abel, at the place already quoted, and again 



* In reality the osculation will be that of two imaginary, not of two real 

 branches of the curve, since the equal values alluded to are imaginary, as 

 will be seen hereafter. 



