TERNARY DIFFERENTIAL EQUATIONS. 105 



NoWj each of the factors equated to zero evidently satis- 

 fies the conditional equation (2). 



8. It is readily seen that (11) is satisfied by (9) the 

 general primitive, and also by R, — R2, = o, an equation 

 which is not included in the general primitive by giving 

 to C any value independent of w, y, z. Therefore R, — 

 Ri = o is a singular solution of (11). 



The following simple rule derived from the above is of 

 some importance in finding the singular solution of a dif- 

 ferential equation from its general primitive. 



Solve the general primitive algebraically with respect to 

 the arbitrary constant, the roots being generally functions 

 of Xj y, z. The condition of equal roots is a singular so- 

 lution. 



Professor Cay ley informs me that the above rule is well 

 known at Cambridge. This may be so ; still I have never 

 seen it in print, and it is not mentioned by Boole in his 

 ' Differential Equations.' It was discovered by me some 

 fifteen years ago, when I communicated it to my pupils 

 and mathematical friends, amongst whom I may mention 

 the names of Rev. Robert Harley, F.R.S. &c. aad the late 

 Mr, Parkis, senior Wrangler of 1864, and Vice-Principal 

 of the School of Naval Architecture at Kensington. It 

 may be further stated that if a difi'erential equation of two 



variables be solved algebraically with respect to ^ ( = -- ), 



the roots being functions of x, y, the condition of equal 

 roots which satisfies the differential eouation is a singular 

 solution. 



