IN THE THEORY OF DIFFERENTIAL EQUATIONS. 



529 



(xy)-curve is a circle on the diameter OP. Any point Q on this circle has its (aft)-curve passing 

 through P: that is, OQP is a right angle. Any point Q on the straight line has a polar 

 circle passing through P. That is, the direct and converse properties of the angle in a semi- 

 circle are only expressions of the reciprocal polar property of x 2 + y 2 = ax + by. Again, if the 

 point (a, b) describe a directing curve, then, since a, b, db:da, determine x, y, dy.dx, the 

 (a?y)-curves produced all touch, with a certain order of contact, a certain connecting curve : if 

 this connecting curve be made a directing curve for (x, y), that which was the directing curve 

 of (a, b) becomes, with the same order of contact, the connecting curve of the (a6)-curves pro- 

 duced. If <p(x, y,a,b) and (p(a,b,x,y) be identical, then for a given pole the two polar 

 curves are identical : and if the pole be taken on the curve <p(x, y, x, y) = 0, the polar curve 

 touches this last curve in the pole. 



1 2. The points (a, b) and (x, y) are not determinants of each other : but a, b, b' (or 

 db-.da) determine x,y,y' (or dy.dx). That is, any element of any curve determines an 

 element of another curve : and these curves are the reciprocal curves of direction and connexion 

 just now described, which may be called reciprocal polar curves*, the modular equation being 

 <p (x, y, a, b) m o. Let this give a = A(x,y,y'), b = B : then db: da or b' = B v ,:A y ,, y" - x 

 disappearing. Accordingly, a, b, b', depend upon x, y, y'. Hence b", or db'-.da, depends upon 

 ?» Vi V\ y" ? an d so on - The method in my last paper, by which every instance of 

 <p(x, y, a, b) = is made a means of transforming any equation of two variables into another of 

 the same order, amounts to a reference of the curves defined by the given equation to their 

 polar reciprocals, upon a modular equation taken at pleasure. Let <p (x, y, a,b)=0 give 

 x = X (a, b, b'), y = Y(a, b, b'), y = Y b ;.X v , &c. : substitute these values in any given equa- 

 tion f(w, y, y', y", &c) = 0. If the result can be integrated into F (a, b) = 0, then x = X and 

 y = Y may be made to express x and y in terms of a ; whence the primitive of / = 6 can be 

 found. This may be called the method of polar transformation^ : under it may be brought 

 nearly all the isolated methods which end in algebraic elimination. 



The modular equation being <p{x,y, a,b) =0, and a curve y =fx being given, the co-ordi- 

 nates a, b of its polar reciprocal are seen in a = A(x,fx,fx) and b = B {x,fx,fx). Two 

 curves, y = fx, n = Fm, being given, any one modular equation, <p (x, y, a, b) = 0, may be 

 made the means of making these curves polar reciprocals, with such an amount of conditions to 

 spare as may suggest the means of solving many problems. Assume m = fx(a,b), n = v (a, b) : 

 and thence a — a (m, n), b = /3 (m, n). We now see that m = n (A, B), n = v (A, B) are 



* Thus the equation x^ + y'^ax + by gives the following 

 theorem ;— If the curve A be touched by the perpendiculars to 

 the radii of B, the curve B is touched by the circles whose 

 diameters are the radii of A. This very obvious theorem is not 

 common : had it been so, surely the following would have been 

 a common exercise for beginners ; — The circle described on any 

 focal radius of a conic section touches the circle described on 

 the major axis. 



+ The most remarkable case, deduced from y = ax— b, may 

 be called linear. When I gave (in 1848) the polar linear 

 transformation for partial differential equations, which I found, 



before the paper was printed, to have been fully given by 

 M. Chasles, I noted that Legendre had made an application 

 to a certain partial biordinal, and that Lacroix had made an 

 imperfect (or all but perfect) transformation of a partial 

 primordinal. I may now add that Ampere, in a paper pre- 

 sently cited, has given more than one instance of linear trans- 

 formation for partial equations, but only with reference to one 

 class of equations, and without any hint of general application. 

 I have as yet found no trace whatever of any thing beyond the 

 linear method even for partial equations, and none of any case 

 whatever for equations of two variables. 



68—2 



