118 
Eliminate (C) from (10) by means of equation (9) and there 
results 
(Ri-E2)2&c. 
dy dRi dR^dz 
dy’dx dx dz dx 
(dK.^ly cZR., cZRocZ^) ,, .v 
1 ^ ^ r’“- = ° • • • ( “ > 
Now, 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 Rj - R 2 equation which is not 
included in the general primitive by giving to (C) any value 
independent of x,y,z. Therefore, Ri-R 2 ^ 0 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 diffe- 
rential equation from its general primitive. 
Solve the general primitive algebraically with respect to 
the arbitrary constant, the roots being generally functions 
of x,y,z. The condition of equal roots is a singular solution. 
Professor Cayley informs me that the above rule is well 
known at Cambridge. This may be so ; stiR, 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 communicatied it to my pupils and ma- 
thematical friends, amongst whom I may mention the names 
of Rev. Robert Harley, F.R.S., &c., and 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 differential equation of two variables be solved 
algebraically with respect to = roots being 
functions of x,y, the condition of equal roots which satisfies 
the differential equation is a singular solution. 
Professor OsBOKNE Reynolds exhibited various forms of 
vortex motion in a large glass tank by means of colour, or 
bubbles of air, the vortex lines behind an oblique vane, 
the vortex ring behind a circular disc, the vortex rings 
