POINTS OF THE INTEGRAL CALCULUS. [113] 



Let P be the point of concourse ; take Q and R infinitely near to it on the two curves, 

 and let the tangents at Q and R meet in T. If the increments of the angles of revolution of 

 the tangents have a finite ratio, that is, if the curvatures be both finite (or both nothings or 

 infinites of the same order) then .ST and QT are of the same order, and both infinitely great 

 compared with QR. Hence the angle RTQ is infinitely great compared with QR, subject, it 

 may be, to exception for some isolated directions of QR : and this angle is d% -=- (1 + % s ) ; d% 

 being formed in the passage from Q to R. Hence ^ x d,v + Xydy is generally infinitely great 

 compared with ^/(dx 2 + dy 2 ), and independently of dx and dy : from which it follows that 



X* or Xy ' s infinite at P< 



Conversely, let there be a curve on which %* or ^ is infinite. Take a point P on this 

 curve which is not of infinite curvature, and if a primitive of y - y can be drawn through this 

 point, take Q and R on the given curve and on the primitive, infinitely near to P. Then if 

 the primitive be also of finite curvature, and if tangents be drawn as before, it follows that, as 

 d-% is infinitely great compared with ^/(dx : + dy 2 ), the angle QTR is infinitely great compared 

 with QR. From this it follows that there is a contact of the first order at least, or that y is 

 the same in both curves, or that y = ^ is satisfied on the first curve. All this requires more 

 amplification, and will not be received without objection except by those who give it. 



We collect then the following theorem, remembering that at a point which has not infinite 

 curvature, y" or x* + Xj/X ' s not infinite. Any relation which satisfies v, = co or y_ y = co 

 certainly satisfies y' = x whenever X* + X»X * s not infinite, and may satisfy y = \ wnen 

 X* + XyX *'* infinite. And in the first case, the curve ^=00 or v = co has contact with 

 all the primitives which meet it, but in the second, whether with or without contact, it is the 

 locus of points of infinite curvature in the primitives (most commonly cusps). But when 

 X x + Xv X * s infinite, it does not follow that y' = ^ is satisfied. 



The following is another proof of the novel part of this theorem. Let the point (x, y) 

 be called the pole of the straight line 



n -y= x {x, y).(g-#). 



This line is tangent to any solution of y = ^, which passes through (.r, y). And in order 

 that a family of straight lines may all touch such a solution, the ultimate intersection of any 

 two which are infinitely near to each other must be the pole of either. Now the intersection 

 of the line above given with that in which ,v, y, v become as + Ax, y + Ay, % + A^. is 

 determined by the equations 



^X ^X 



So that, when the ultimate intersection is the pole itself, we must have 



to - 



= 0; 



X*+X* m 

 m being the limiting ratio of Ay to Ax. 



If P and Q be the two contiguous points, and T the ultimate intersection of their polar 

 lines, we now see as follows. If P be not on the curve v, = co or ^ = co, there is only 

 Vol. IX. Pakt II. 39 



