and the Theory of Conjugate Poijits. 95 



keeping in mind the additional principle, that when y becomes 



(111 

 imaginar}', for a particular value of x, the coefficients -j-, 



-7-^, -r^^ &c. must also be imaginary for the same value pf*«'> 



inasmuch as differentiation does not interfere with the radi- 

 cals which affect x in the original function y. We thus ob- 

 tain the following criteria: — •^j'Tr/ti'nJ ? 



1. Let the equation be solved for one of the Variables ; then 

 if any pair of values for .r and y which satisfy the resulting 

 explicit equation, difller from real values only by the sym- 

 bol \/ — 1, in whichever direction we arrive at the zero, those 

 real values will be the co-ordinates of a conjugate point, (ij^np 



2. If it be inconvenient to solve the equation for one of the 

 variables, we may take a differential coefficient of 4iny order 

 whatever; then, if any pair of values for x and 7j render this 

 coefficient imaginary, and at the same time satisfy the rational 

 equation of the curve, those values will be the co-ordinates of a 

 conjugate point; it being observed, as before, that if the ima- 

 ginary value differ from a real value only by OV'— 1, this 

 zero must remain imaginary in whichever direction we arrive 

 at it. In the contrary case, a limit or a cusp will be implied, 

 and not a detached point. ' i^.-oci u 



Let us apply these principles td*^ example already quoteid' 

 from Mr. Gregory, viz. -rfj ,t!>70v.'O! ' >! jfli 



{c^y-:^f = Ix-hf ix-a)%"a<-bi '-" " "V'^^'tl 



solving this for j/, we have ; gi? ijaw zn 



d^y = {x-bf {x •a)3 + ^, v/od ion- 



which, for x = a, gives «/ = ^ ± 0\/— m^ (my^hig"p'tit f6^' 



A— a) ; and as the zero remains imaginary, whether .«• = a be 



arrived at from a succeeding or a preceding value of ,r, we 



a^ 

 conclude at once that the co-ordinates x = a^ y z= —^^ belong 



to a conjugate point. 



If differentiation be employed, we find the imaginary to be 



d v d^ V . . d? y 



zero, both in -f^ and in -~,^ but to be finite in -~. We 

 dx dx^ dar 



may therefore conclude that in the imaginary curve through 

 the conjugate point, y, -^ and -r-^ each give equal imaginary 



(p y 



values, while the imaginary values of -j^ ^**e unequal a^. that 



