188 MR. ROBERT RAWSON ON SINGULAR 



Let 



(|) i -^ + y +I )2- 2a?(a? - y " l)=0 ^ ■ (44) 

 whose complete primitive is 

 c x — c{x + a? % -y— log (x—y)} + (a? -y){x— log {x—y)} = o. . (45) 



Now x—y=o is a solution of (44) ; is it a singular or a 



particular solution? The primitive can be put in the 



form 



{c-x + log(x — y)}{c + y-x z } = o. . . (46) 



Therefore 



c— <2?+log {x—y) = 0, 



from which 



e x 

 x—y = — =o ; when c= infinity. 



€ 



Hence x—y = o is a particular solution, the constant (c) 

 being infinite. It remains to eliminate (y) from (44) and 

 (45) D y nieans of 



w = x-y (47) 



Then 



fdu 

 \dx 



V dw 



-J +(2jj-fi;~l)— _ W (2ff-l)=o, . (48) 



c* — c{x z + w — logw} + (x z — x + w)(x— \ogw)=o. . (49) 



Now w = o is a particular solution, and satisfies (48), 

 but the primitive (49) still remains a function of x, which 

 is in opposition to the principle used by Boole (supp. vol. 

 p. 28). 



14. Boole's proof of a theorem from Euler. 



First eliminate (y) from the given differential equation, 



and its complete primitive by the relation w = (f>(x, y) ; 



then 



dw , , N 



3£=^foM0, (50) 



