86 Mr. H. S. Warner on Conjugate Points. 



ginary */ — 1 does not disappear, and its coefficient B has a 

 value, whether positive or negative, different from 0, then the 

 curve is unassignable on this side of the ordinate at which 

 B = as well as on the other ; and hence the equation ex- 

 presses at this portion of the locus a single point, isolated and 

 unconnected with any (assignable or real) part of the curve 

 that is a conjugate point. 



6. Hence the following rule : — Having put the equation 

 under the form y =f } (%) +J^ (x), then if for any particular 

 value of x such as a',J^(x) becomes of the form A-fO'/ — 1, 

 while both for a value of % a little greater and for one a little 

 less than a' (such as a' + h) it is of the form A + B V — 1, there 

 will be a conjugate point determined by the co-ordinates 

 x = a',y ==/i (a') + A ; but if/* 2 (x) is of this form only on one 

 side of a', there will be a cusp determined by these co-ordi- 



•nates. 



In the case of an implicit equation : if any values of x and 

 y, such as #=«', y=b', satisfy the equation, while both when 

 x is taken a little greater and a little less than a', y must be of 

 the form A + B*/— 1 ; to satisfy the equation there will be a 

 conjugate point at x=a' f y=b'; but there will be a cusp 

 instead of a conjugate point if it is only on one side of a!, that 

 y is of the form A + Bv / -1. 



7. In dividing <$ (x) into the two other functions, f x (x), 

 f<t (x), we must take care so to assume^ (x) that it may be pos- 

 sible for all the parts of the locus that we wish to examine in 

 regard to conjugate points. If <$> (x) cannot be divided into 

 two functions such as we desire, we can always obtain the ne- 

 cessary form of equation y=J\ {x) +f 2 (x) by taking/, (r) = 0, 

 andj^(x) = <$>(x), so that the equation is y = 0+ <p (#), and the 

 axial curve or locus of y =/, (x) is the locus of y = 0, that is, 

 a straight line coinciding with the axis of x. 



8. As an example, let us take the equation 



\a—x) 



2 fir, 



whence we have y = $ + (a — x) Vx — 2a; 

 then f x (x) = x, /% (x) = (a — x) Vx — 2 a. 



Now when x = a, / 2 {x) — 0\/—a=zOV—l', 

 while for x — a + h (k being less than a) it is equal to 



+ h »/a±h . V - 1 ; 

 hence there is a conjugate point determined by 



* = «> y-f\ («) = «. 



