532 Ma DE MORGAN, ON SOME POINTS 



For the singular solutions we have 



4>xy + x* x 2 x /l6y + 4a? 2 + x* 

 ~4(l+a? 2 ) = 4 4^ 1+x* 



4«/' - x 3 x 1 /l6y + 4a? 2 + at 



= 4(l + a? 2 ) = " 2 + 4 V i +0? 2 ' 



aj=-b'=2{b + v / (b* + a)}, 



y = ab" - (2ab + b) b' + 6 2 - a 2 = 12a6 2 + 3 (a 2 + 6 2 ) + (l2a& + 2&) v /(6 2 + a 2 ), 



the primitive of Q = (obtained by linear polar transformation), is 



2 (b + \Zb' + a) . y/(l + 4a + 8& 8 - 86 \A 8 + a) 

 + log {2 (6 - y/W + a) + y/[\ + 4a + 8& 2 - 86 \A 2 + a) } = Z. 

 Substitute in this the values of a and b, and we have for the primitive of P = 0, 

 2 p . 1 ^16^+4^ ^ + ^ + ^ {v/(l+ , 2) _^ =C; 



as otherwise found by Lagrange. 



The singular primitive of P=0 is l6y + 4>a? + a? 4 = 0, which may be obtained from 

 (h = 0, d> o = 0, (p b = 0: it does not solve y"= ^. There is no finite singular solution of 

 Q = 0. The relation 1 + a? = happens to be a singular solution of y" = ^. 



By choosing instances in which the multipliers of constants are functions of only one 

 of the variables, we are obliged to adopt cases of considerable complication, if we would 

 exhibit all the phenomena. If we take as our example 



a*y — axy + mx = b, with a s y + mx — 2ma = 



as the condition of singular solution, we find for the biordinals 



(m + xy) y" 2 + 2y {y - xy) y" - 4>y s = 0, a s 6" 2 - 2 (2ma + ab'- b) b"+ 4wi6 = 0, 



for the singular primordinals and singular primitives 



(xy'+ y)*- Irny = 0, (2ma + ab'- 6) a - 4ma 2 6'= 0, 



C=\os,\— ^ — J ~ V » 6 = ma U +(Z + log a) 2 L 



\ x ) v m 



in which C = Z + log — — . The singular solutions of the singular primitives are 



y = o, x = 0, a = 0, b = ma. 



14. The substitution of compensating variables in place of constants is the key to many 

 of the general properties of differential equations. What effect this substitution shall have, as 

 to alteration in extent of solution, depends upon the number of constants in relation to the 

 number of equations of compensation : and this effect may consist in diminution, permanence, 



