542 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



We have (article 27) : 



A2v = Av 2 + 2Av . Av . 



A2v = Av 2 + 2A^ . Av , 



A2v = A^ 2 + 2A^ . Av . 



whence 



A2v - A2v = (Aw - Av) (Am - 2 Av + Av) = 



so that the two curves A and A intersect each other on the ordinate at 2v ; in 

 other words, v must be the half of T. From this latter consideration it follows 

 that, in tabulating the intersections, it would be enough to carry the computa- 

 tions as far as to JT, just as in the construction of the trigonometrical canon it is 

 sufficient to make the calculations up to half a right angle. 



Fig. 3. 



42. The relative positions of these successive intersections may be clearly ex- 

 hibited by the following artifice. Let us suppose that on one of the ordinates, as 

 that at u, figure 3, a plane is set perpendicular to the plane of the paper, and 

 that through the points u, 2, 1, lines are drawn normal to the picture; 

 these lines being represented generally by the horizontal lines of figure 4. 

 This arrangement being made for each successive ordinate, let an equilateral 

 trigon ABC (fig. 4) be constructed so that its corners may be upon the 



