[120] PROFESSOR DE MORGAN ON SOME 



wholly unconnected. If then one of the constants C B , &c. be substituted in terms of the rest 

 from /= 0, we have the complete integral of the equation. By differentiating /= 0, we obtain 



(P (px + P, (p,x + ... ) y (n * l) = 0, 

 P, P„ &c. being partial differential coefficients of/ with respect to V m >n , &c. and (px, rp t x, &c. 

 being the several rational functions of these last. By making the first factor vanish, and 

 eliminating y {n) between this and/= 0, we get an equation of the (n - l) th order, the ordinary 

 primitives of which are singular solutions of/= 0. 

 For instance, let the equation be 



r 2 



w n i / n i\ it 



— y -xy'+y = {xy -y)y . 

 2 



Here y = C + C L x + C 2 x i , gives C Q = - 2dC 2 , and the complete primitive is 



y= -2CA+ d»+ C 2 x\ 



Differentiating the equation, and throwing out the factor y" , we have ^a? 2 = xy" — y + xy", 

 and elimination of y" gives 



4y' 2 - 12/rV + l6xy + a? 4 = 0. 



Of this y = -j-g- 0? is an ordinary solution, and a singular solution of the original : but 

 y m 1 a? is a singular solution of the last, and is not a solution of the original. 



Let/(F m n , V m - „_,) = 0. The preceding method gives only n — 1 constants, or n constants 

 with one relation between them. But if we write % for y w the preceding takes the form 



f(f(bx.ss'dx, jyfyx. ssdx) = 0. 



If we differentiate with respect to x, and eliminate ftp,v . x'dx, we have an equation 

 of the form 



\J/ (x, Z, f^rX . zdx) = 0, 



which is of the first order. When z or y w is found from this, y is found by integration, but 

 we must take care to introduce the relation known to exist among the n constants which the 

 integration brings in. 



In the same manner f(V mn , V m ' >n _ v F m » )B _ 2 ) = is reducible to an equation of the second 

 order : and so on. 



If Vm,. = ft* • dy w and V m . >n _, = fya, . dy^\ we have 



m ' n J dx\^/X dx } J 



Whence it follows that all differential coefficients of V mn can be expressed in terms of 

 those of V m - >n _ l , and thence in terms of those of F OTj „_ 2 , and so on. If then a differ- 

 ential equation contain x, V mn , and differential coefficients only of V m - ftt+v V m " iK+2 , &c, 

 the substitution of « for V m „ will reduce the equation n units of order. 



The following examples will illustrate a large class of transformations, one of which is that 

 which corresponds, in equations of one variable, to M. Chasles's method for the case of two. 



According as the rational function is (px or \|/#, let0„or \^„be used to represent V m>n . Let 



