186 MR. ROBERT RAWSON ON SINGULAR 



z iA i w i) ma 7 be added' to the primitive (22), thereby giving 

 rise to other singular solutions, w l = h l &c. • 



12. The second problem is more difficult to answer 

 than the first, and seems to have been first generally 

 considered by Laplace himself. (See Boole's Diff. Eqs. 

 p. 174, 2nd ed.) 



Let 



J +r=0 (39) 



be the differential equation whose singular solution is 

 required, (r) being a function of x, y. 



By reference to equation (21) f(w), being the special 

 object of inquiry, is the factor which makes (2 1) an exact 

 differential equation, therefore (39) admits of the form 



dy f(w)r , x 



i + TW =0 (+0) 



If, therefore, (40) is an exact differential equation, then 



or 



Since 



dp_ f"(w) f dw dw\ 



dy f(w)\ dy dxj ' ' ' ^ ' 



, dp dr 



p=—r and -f- = r . 



dy dy 



Now by paragraph (7) the singular solution w — h = o 

 makes f(h) = infinity, therefore /' (w) must, for a singular 

 solution, necessarily be of the form 



/w-S m 



whereof the function </> (w) does not vanish for w = h. 



